Rank Eaualities Related to Generalized Inverses of Matrices and Their Applications

Yongge Tian

in

The Department

of

Mathematics and Statistics

Presented in Partial Fulfillment of the Recluirements for the Degree of

Master of Science at

Concordia University

Montréal, Québec, Canada

April 1999

@ Yongge Tian, 1999

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ACKNOWLEDGEMENTS

1 wish to express my deepest gratitude to my thesis advisor Professor George P. H. Styim for his guidance.

iritduable discussions and thorough understanding during the preparation of this tlwsis.

1 would d s o like to express my sincere thanks to my supervisor professor Ronald J. Stern for his

advice and generous support in preparing this thesis.

My Iieartest thanks go to Professor Zohel S. Khalil and Professor Tariq X. Srivastava for their deep

concern and continuous encouragement at carious stages of the programme.

1 mi d s o cluite grateful to Dr. Siao-wen Chang and Professor Yoshio Takarie for givirig ~ ~ l i i a b l c

mmrnerits and suggestions.

This thesis is a surnrn~zry of the work 1 accomplished in the pcxt several years at Zhengzhou Technicai

School for Surveying and Mapping, Zhengzhou, China. and then a t Concordia University. The wIioIe

n-ork was mostlÿ motivated by an earIier sernind paper of 'vlarsaglia and Styan on rank eqiialities arid

irieqimlities of matrices published in Linear and SZultilinear Algebra 2(1074): 269-292. 1 \vas attracteti

l>y the heauty and usefuiness of the results in that paper. and cleepl- believeci that theu wciiild c-crtainiy

I)t:come a powerful tool for clealing mith various problems in the thcory of generalizeci i~iverses of rriatrices

aritf its applications. Tlirough several years' effort. 1 clevelop the results in the lliirsaglia and Styaris'

paper into a complete mettiod for establishing various equalities for rCnks and gcneralizecl inverses of

rriatrices. Parts of the thesis were published in [91], [92], [93]: [O-l]. Otlicr parts (as a joint paper \vitil

George P. A. Swan) were presentecl a t the Seventh International CVorlishop on .\latriccs ancl Statistirs

(Fort Laurlerdale. Florida. December 11-14, 1998). In addition. sorrie parts of the tiiesis hiivri rccentiy

heeri siibnlittecl for publication. They are Chapters 3. 4. 5 and Il .

L u t but not the least? 1 am greatly indebted to rny wife Hua Gan and rny son Nigel Zlie Tiari for

their patience, uncierstanding, and encouragement during my studies in China and Canada.

ABSTRACT

Rank Equalities ReIated to Generalized Inverses of Matrices and Their Applications

by

Yongge Tian

This tliesis develops a general method for expressing ranks of niatris cxpressions that involve the lloorc-

Pcrirose inverse. the group inverse. the Drazin inverse, as welI as the weighted Moore-Penrose iriverse of

rriatriccs. Through this method we estabiish a variety of valuable rank equalities related to generalizcd

inverses of matrices mentioned above. Using them. we characterize many matrix equaiities in the theory

of gcneralizecl inverses of matrices anci their applications.

Contents

.................................................................. 1 . Introduction and preiiminaries (1)

............................................................................. 2 . Basic rank formulas ( 5 )

3 . Rank equdities related to idempotent matrices and tfieir applications .......................... (10)

.................. 4 . 1Iore ori rank eclualities relatcd to idempotent matrices and tlicir applications 121)

5 . Rank equalities related to outer inverses of matrices ............................................ (33)

.............................. G . Rank eclualities related to a rnatrix and its Moore-Penrose inverse (-LO)

. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Rank eclualities related to matrices and their Moore-Perirose inverses ( 4 1 1

.......................... S . Reverse order laws for Moore-Penrose inverses of products of matrices ((35)

9 . lloore-Penrose inverses of block matrices ...................................................... (76)

........................... LO . Rank eqrialitics relatecl to Moore-Penrose inverses surns of matrices (92)

............................................ 11 . lIoore-Penrose inverse of block circulant matrices ( LOO)

.................................. I2 . Rrulks equalities for submatriccs in kloore-Penrose inverses ( 107)

.................................................... 13 . Rank equalities relateci to Drazin inverses 1 13 1

.................................... 14 . Rank equditics for submâtrices in Drazin inverses ... .... (125)

15 . Reverse order iaws for Drazin inverses of products of matrices ............................... ( 129)

.................................. 1G . Rank eclualities related to weighted Moore-Penrose inverses ( 133)

17 . Reverse order laws for weighted Moore-Penrose inverses of products of matrices .............. ( 140)

.................................................................................... Bitdiog~aphy ( 1.16)

Chapter 1

Introduction and preliminaries

This is a cornprchensive work on ranks of matri.. expressions involving Moore-Penrose inverses. grotip

iriverses, Drazin inverses, as well as weighted Moore-Penrose inverses. In the ttieory of genernlizecl invcw~s

of niatrices and ttieir applications. there are numerous matris expressions and ecliialities that involw tticsix

tliree kinds of generalized inverses of matrices. Non- we propose such a problern: Let p( -4:. - - - . -4; 1 ;~rid

q( 0:. - - - . B: ) be two rnatrix expressions involving Moore-Penrose inverses of marrices. Tlieii rleterriiirir

necessay and sufficient conditions sucli that p( -4:. - - - . -4: ) = q( B:. - - - . B: ) holcis. -A srerningly tririal

c-unclition for t h equality to Liold is apparently

However, if n-e can reasonably find a formula for espressing the rank of the left-hanci side of LI-( 1.1)-

t h n we can clerive imrnediately from Eq.(l . l) nontrivia1 coriciitions for

to Iiold. This work h a . a far-reaching influence to many problems in the tlieory of generalized inverses o f

rriatrices and their applications. This consideration motivates us to rnake a thorough investigation to t h

n-ork. Iri Eict. tlie author lias successfully used this idea to establish necessary and siifficient coriditioris t t siicli that (-4BC)t = Ct Bt-4t. (-ABC)' = (BC)tB(.4B)t, and (rli.4- - - - - 4 k ) t = -4;. - + + -A2--LL (cf. Tim.

L992b. 1994). But the rnethods used in those papers are somewhat restricted and not applicable to \-ririoris

kintl rriatris expressions. In tliis thesis. we shall develop a general and coinpletc rtiethotl for cstablisliirig

rirrik eqiialities for matris expressions involving Moore-Penrose inverses. group inverses. Drazin inverses.

a s ive11 as weightecl Moore-Penrose inverses of matrices. Using these rank formulas. wc shall cliaractcrize

various equalities for generalized inverses of matrices. anci then preserit their applications i n tiic tiicwry

c ~ f gcneralized inverses of matrices.

Tlie niatrices considered in this paper are al1 over the cornples number field Ç. Let -4 E Cr'" ". \\-P

use -4'. 1-(-4) ancl R(,-I) to stand for the conjiigate transpose, tlie rank and the rarige (column space) of

-4, respectivelj-.

It, is weIl known that the Moore-Penrose inverse of matris -4 is defined to be the unique solirtio~i .Y of the following four Penrose eqiiations

and is often denoted by ,Y = ~ t . h addition. a matris S that satisfies the first equation abovc is calleci

an inner inverse of -4. and often denoted by A-. A matris S that satisfies the seconcl equatiori abovc is

calleci an outer inverse of -4, and often denoted by ,-l('). For sirnplicity, we use E.4 anci FA to starid for

the two projectors

E = f - 4 t and F A = f - . 4 t - 4

inctuced by -4. -1s to various basic properties concerning Moore-Penrose inverses of matrices. see. e.g..

Ben-Israel and Grevilie (IWO), Campbel1 and Meyer (1991). Rao and 1,litx-a (1972).

Let -4 E C r r i X n ' be given tt-ith Ind-4 = k, the srnallest positive integer such t h t r(.4k") = r(-4') .

The Drazin inverse of matris -4 is defined to be the unique solution S of the following threc equatioris

(1) * Y = 4 (2) X.4S = S. (3) ;Lx = S A ,

itiid is ofteri denoteci bu -Y = --lD. In particular, when Ind -4 = 1. the Drazin inverse of m a t r k -4 is cal1

the group inverse of -4. and is often denoted by -4*.

Let -4 E Cn' ". The weighted Moore-Penrose inverse of -4 E C"'" " with respect to the ttw positivc

clefinite matrices 44 E C n L X r n and N E C n X n is defined to be the imiqiie solution of the follov-ing four

niatrilc equatioris

t and this S is often clenoted by S = AL^,^^. In particulnr, when 111 = 1,. irnd Ar = 1,. ;ldLr.,v is the

(:onventional Moore-Perirose inverse -4t of -4. Various basic properties concerning Drazin inverses. grotip

iiiverses and LVeiglitetl Lloore-Penrose inverses of matrices can Le found in Ben-IsraeI and Grevillc ( 1980).

Canipbcll and Meyer [1991), R.ao and hlitra (1971).

It is ~vell known that generalizecl inverses of matrices are a powerful tool for cstabIis1iing uriot ls r m k

cqrralities on matrices. -4 seminal reference is the paper [Xi] by hlarsagliü anci S cyan (1974)- 111 tliat

piper, some fundamental rank equaiities and inequalities related to generaIized inverses of matrices \ver(?

estltbiislied and a variety of consequences and applications of tliese rank cqualities and iriequalities were

ccinsidered. Since then, the main results in that paper have widely been applying to deaiing with vxious

protilerns in the theory of generalized inverses of matrices and its applications. To some estent. this ttiesis

cciiilcl be rcgarcted as a summary and estension of ail work related to that rernarbble paper.

WC nest list some key results in that paper. ~t-tiich will be intensively appliecl in this thesis.

Lemma 1.1 (Marsaglia and Styan, 1974). Let -4 E C''Lx". B c m x - E C ' [ X I L ILTLII D E C"'. T h 1

where S.4 = D - C-4 B is the Schur cornpiement of -4 in hi = / 1 , m i

crrlled the rnnk complement of D in M. In particular. if

R(B) Ç R(,4) and R(C*) C R(-4').

The sic runk epcrlities in Eqs.(1.2) -(1.7) are also t m e when replacing -4t b y -4-.

Lemma 1.2 (Marsaglia and Styan. 1974). Let -4 E C'"lx ". B E C ~f clX" und D E C l x k . Th:n

(a) r [ -4. B J = 4-4) + r ( B ) R(=l) n R ( B ) = ( 0 ) R[(E,\ B) '] = RIB* ) R[( E B . ~ ) ' ] =

Lemma 1.3 (Marsaglia aiid Styan. 1974)(rank cui~cellrrtion n ~ l e s ) . Lct -4 E C n L X ". B E mrd

C E L"" he given, and suppose that

( 4 ) = ( 4 ) and R[(P.4)'] = R(-4").

Tlr en

Lemma 1.4 (Marsaglia and Styan, 1974). Let A, B E CmXn- Then

L J

( c ) If R(,4") n R(B-) = {O), then r ( -4 + B ) = r[,-l. B 1. (c i ) r-( -4 + B ) = r (A) + r (B) e r( -4 - B ) = 4.4) + r ( B ) R(-4) n R(B) = {O) , und R(-4') n

R ( B S ) = { O ) .

In acldition, ive shall also use in the sequel the folloming several basic rarik formulas. which are either

well kriow-n or easy to prove.

Lemma 1.5. Let -4 E C m x n . 8 f C n x m und IV E CWLxm. Then

Lemma 1.6. Let -4. B f C m X n . Then

Pro OF. FoIIows from the following decomposition

Lemma 1.7 (Anderson and Styan, 1932). Let -4 f CnLXm. Then

This thesis is divided into 17 ctiapters with over 240 theorenis and corolIaries. TIicy orgariize as

folIotvs.

In Chapter 2. we establish several universal rank formulas for matris espressions tliat irivoive lloore-

Pcrirose inverses of niatrices. These rank formulas will serve 'as a basic tool for developing the content in

a11 the subsequent sections.

In Chapter 3, ive present a set of rarik formulas related to surns, clifferences and proclucts of idempotent

triatrices. Based on them, we shali reved a series of new and untrivial properties related iclernpotenc

rliiltrices.

In Chapter 4, ti-e extend the results in Chapter 3 to some niatrix expressions that involve both

idempotent matrices and general matrices. In addition, we shd l also establish a group of new rank

formulcas relateci to involutory matrices and then consider their consequences.

Iri Cliapter 5. we establish a set of rank formulas related to outer inverses of a niatris. Sorne of them

t d l be applied in the subsequent chapters.

Iri Chapter 6: we examine various relationships between a rnatri,~ and its LIoorc-Pcrirose irlvrrsc using

the rank cqualities obtained in the preceding cliapters. UYe also consider in ciie ckaptc!r lion- c-h;ir;ictctrizc~

sorrie special types of niatrices, such as. EP macris. conjugatc EP niatrîx. Ili-EP rriatris. star-tfaggcv-

niatris. power-EP matris, and so on.

In CIiapter 7. we discuss various rank equaLities for m a t r k expressions that involvc, r.wo or more hloorv-

Penrose inverses. and then use them to characterize various rnatrk equalities tliat i~ivolve 3loore-Perirose

iriverses.

hi Chapter S. we investigate various kind of reverse order laws for Moore-Perirosc inverses of products

of tlb-O or thrce niatrices using the ranlc equalities established iri tlie preceding chapters.

In Chapter 9. me investigate Moore-Penrose inverses of 2 x 2 block rriatrices. as tvell as I L x ri. i~lock

niatrices using the rank equdities established in the preceding chapters.

In CIiapter 10. we investigate Moore-Penrose inverses of sums of matrices using tlie rank eclualities

established in the preccding chapters.

In Cliapter 11, ive study the relationships between Moore-Penrose inverses of block circulant rriatrices

and sunis of matrices. Based on them and the results in Cliapter 9. ive shall present a group of csprcssions

for hloore-Penrose inverses of sums of matrices.

1x1 Chapter 12. we present a groiip of formulas for espressing ranks of subniatrices in the lloorc-Periroscx

inverse of a n i a t r k

In Chitpters 13-17: our work is concerned with rank equdities for Drazin inverses. group inverses. a ~ i d

weigtited Moore-Penrose inverses of matrices and their applications. Various kincls of problenis exanlined

in Cliapters 6-12 for Moore-Penrose inverses of matrices are alrnost considered iri tliese five dmpters for

Drazin inverses. group inverses. and weighted Moore-Penrosc inverses of matrices.

Chapter 2

Basic rank formulas

The first and most fundamental rank formula used in the sequel is given beloiv-

Theorem 2.1. Let -4 E L n L X n , B E C m X " : C E C l X crnd D E CLX"c giuen. Then the runk of thri

Schw complement Sti = D - C-4tB satzsfies the eqvality

Proof. It is obvious that

R(.-I'B) C R(.4') = R(-4*-4-4*)t and R(,4Cœ) Ç R(=L) = R(,4,4*.-1).

Tlicrr it follows by Eq.(1.7) and a well-known basic property -4*(-4*.4-4')tA4' = --Lt(see Rao arid JIitra.

1971, p. 69) that

T h significance of Eq.(fZ.I) is in that the rank of the Schur corripIcrrient S.-\ = D - C-4tB cari Lie

cvaliiüted by a block matris formed by -4. B. C arid D in ic. wtiere no restrictions are imposeci or1 S,-L

;LILCL 110 Moore-Penrose inverses appear in the right-hand side of Eq.(2.1). Ttitis Eq.(2.l) iri k t provides

u s a powerfui tool to express ranks of matrlu expressions that involve kloore-Perirose iriverses of niatriccs.

Eq.(Z.l) cari be cstended to various general formulas. We nest present sorric of theni. whicli n-il1

n-iclely be used in the sequel.

Theorem 2.2. Let -4, : .4?, Bi, Bi, CL , C2 and D ure matrices sucfi tfrat eqr -as ion D - Ci -4; B, -

~ ~ - 4 : B? is defirred Then

Proof. Let

C = [C l r C:!]:

Theri Eq.(Z.l) cari be written as Eq.(2.2). and Eq.(2.3) follows froni Eq.(L.G). 0

If the matrices in Eq-(2.2) satisfy certain conditions. the block matr is in Ec1.(2.2) cari ezisily be reduced

to sonle sirnpler forms. Below are some of them.

CoroIlary 2.3. Let -4 E Cmxn, B E C m x k ? C E CL"" and iV E ckX' be giuen. Then

Theorem 2.4. Let BLi Bt. CL( t = 1. '2! - - - . li) und D are mtrtrices S I L C I L thut ~ ~ T C . ~ S Z O I L D -

c ~ - ~ ! B ~ - - - - - C k . 4 i ~ k is deJîned. Tlien

Theorem 2.5. Let A. B, C. D. P und Q are rr~utrices such that cxprcssion D - CPT.-lCJtB Zs debrrcd.

P*.4Q* P'PP' O

WQQ* O ( 2 .s) O CP* -D

Proof- Note chat

~ ( D - c P ~ , ~ Q ~ B ) = T - Q O B

1 0 C - D

.-lpplyirig Eq.(2.1) t o it and then simplifying yields Eq.(2.S). Eq.(2.9) is derivecl from Eqs.(?.S) bj- thc

rank caricellation tan- Eq.(l.S). O

- r ( P ) - r ( Q ) -

Theorern 2.6. S t r p p s e t l iüt the matr iz expression S = D - cl P [ . ~ ~ Q : BI - c ~ P , ' - ~ ~ Q ~ & is dcfined.

Tf1e.r~

D8DD' O O D'

O P*_-lQ* P*PP* O

O &'QQœ O Q'B

D* O CP' O -

- r ( P ) - r ( Q ) - 1 - ( D ) . ( 2 . L'2)

and then applying Ecl.(2.S) to it produce Eqs.(2,lO). Eq.(2.21) is derived from Eqs.(2-10) by the rirnk

c:nncellntion law Ecl.(l.S). Ec1.(2.12) is a special case of Eq.(2.10). @

It is easy to see that a generd rank formula for

(::Ln also be established by the sirnila method for deriving Eq.('3.10). -1s to sonie cjttier generd ria tris

cspressions, such as

Sk = .40~~-4~P.j .42 - - - P,I;I*.

: L I L C ~ tkieir linear conibinations. the forrnuias For expressing their ranks car1 also bc establisliecl. H o w w r

they are quite teclious iri forrn. ive do riot intend to give them here.

Chapter 3

Rank equalit ies related to idempotent mat rices

A square m a t r k -4 is said to be idempotent if -4- = -4. If we consider it as a matris equation -4' = -4.

then its general solution can be written as -4 = C - ( ~ ' - ) ~ V . where Il' is an arbitrary square niatris. TIiis

assertion cnn easily be verified. In hc t . -4 = I;(V2)tC< appareritiy satisfieç -4' = -4. Xorr- for ariy niatris

-4 witli -4' = -4- rve Iet = -4- T'tien L-(V2)t\- = .4(-4')T-4 = -4-4t-4 = -4- T h ~ s -4 = \ - ( \ - ' ) r \ - is

intIeed the general solution the idempotent equation -4' = -4. This fact clearly implies that any matris

expression that involves ictempotent matrices couId be regarded as cz coriveritional rnatris espression

ttiat involves Moore-Penroses inverses of matrices. Tlius the formulas in Chapter 2 are al1 app1ir:ablc to

tlctermine ranks of matriv expressions that involve idempotent matrices. However because of speciality of

ickrnpotent matrices, the rank equaiities related to idempotent matrices can also bc dedured by ~ i r i o ~ i s

elenieritary methocls. The results in the chapter are originally derived by the rank formulas in Chapter

2. we later also fincl some elementary methods to establish them. So we only stiort- tlicse results in thesr.

c!lerrieritar>- rrriethods.

Theorern 3.1. Let P7 Q E CITLX "' be tu10 idempotent mutrices. Then the difle7rnce P - C) satisfies t h !

rltnk equulities

tliat L P Q O 1

-P O O

O P - Q

Ori thc otlier hancl. note that P' = P and Q' = Q. It is also easy to find by blocl; elerxrentary opcrat.ions

of matrices that

r ( M ) = r

-P O P O O P [-:' O J = . [ O ( ) Q

P Q O

the above two equalities ÿields Eq.(3.l). Consequently applying Eqs. (1.2) and (1.3) CO [ P. Q ]

in Eq.(Xl) respectively yields

Putting Eqs(3.4) and (3.7) in Eq.(3-1) produces Eq.(3.2), putting Eqs.(3.5) and (3.6) in Ec1.(3.1) protliiccs

Ec1.(3.3). O

Corollary 3.2. Let P. Q E C"'""' bc two idempotent matrices. T f ~ e n

(a) R( P - PQ ) n R( PQ - Q ) = {O) und R[( P - PQ )* ] n RE( P Q - Q )"I = {O).

(11) R ( P - C ) P ) n R ( Q P - Q ) = (O) und R [ ( P - Q P ) ' J n R [ ( Q P - Q ) ' ] = { O ) .

(c:) If PQ = O 07- Q P = O ? then T( P-Q ) = r(P)+r(Q), Le.. R(P)nR(Q) = { U ) rrnd R(P')nR(I)* ) =

{ O ) -

(cl) If 60th P and Q are Hermitiun idempotent. then r( P - Q ) = 27'[ P. C) ] - r- (P) - r ( Ç ) ) .

Proof. Parts (a) and (b) follonrs from applying Lemma l.-l(d) to Ecp(3.2) and (3.3). Part ( c ) is a tlircct

cotisequerice of Eqs.(3.2) and (3.3). Part (ci) foIlows from Eq.(3.1). EI

On the bcasis of Eq.(3.1), we can easily deduce a known result due to Hartwig arici Styan (1987) on

the rank subtractivity two idempotent matrices.

Corollary 3.3. Let P. Q E C m X m Oe tzuo idempotent matrices. T f ~ e n the follo~irzr~g . s t ( ~ t ~ ! r n ~ : ~ ~ t . s tz7-t:

cr~i~iualent:

(a) r - ( P - Q ) = r(P) - r ( Q ) . Le., Q sr, P. r -.

(c) R(Q) C R(P) und R(QW) C R(P*).

( c l ) PQ = QP = (S.

Proof. The equivdence of (a) and (b) folIows imrnediately from appIying Eq.(3.1). The qui\-derice of

(b). ( c ) : (d) and (e) can trividly be verified. 0

Iri addition, from Eq.(3.1), we can irnrnediately find the follon-ing result. mhich has been giwn in [3G]

by Gross and Trenkier.

Corollary 3.4. Let P: Q E CniXm be two idempotent matn'ces. Then the following statements c m

(a) The differerrce P - Q is nonsingular.

Proof. Follows clirectiy from Eq.(3.1). O

Xotice that if a 1riatri.x P is idempotent, the In, - P is also idempotent. Tlius replxing P in Eq.(3.1)

1)y Il,, - P. ive get the following.

T h e o r e m 3.5. Let P. Q E CJnX"' be two idempotent matrices. Then the rcrnk of Il,, - P - Q scttisfies

thc equnlities

Proof. Replacing P in Eq.(3.1) by 1, - P yields

(:Mj

(.:3.9j

( 3 . LU)

It follows L>y Eqs.(1.2) and (1.3) that

Putting thcm in Eq.(3.11) proctuces Eq.(3.8). 011 the other lland. replacing P in Eqs.(3.2) and (3.3) 1)'.

Ir,, - P produces

hoth of which are esactly Eqs(3.9) and (3.10). f7

Corollary 3.6. Let P? Q E C n L X n L be two idempotent rnatrzces. Then

(a) R(In, - P - Q + PQ ) n R(PQ) = {O) and R[(i, , - P - Q t PQ) ' ] n R[(PQ) ' ] = {O).

(11) R( I,,, - P - Q + Q P ) n R ( Q P ) = { O ) and R[( In, - P - Q + Q P ) ' ] n R[(C)P)*] = (0).

( c ) P + (2 = Im PQ = QP = O ancl R(P) R(Q) = R(P') - R(Q*) = C"'.

(cl) If PQ = QP = O. then r( In, - P - Q ) = 7 r ~ - r ( P ) - I'(Q).

(c) Irrl - P - Q is nonsingular if und onlg if T(PQ) = r ( Q P ) = r ( P ) = r(C)).

(f) If b o t f ~ P und Q are Hermitiun idempotent, then r( I,,, - P - C) ) = 2r(PQf - r ( P ) - r ( Q ) i I:L.

Proof- Parts (a) and (b) follow from applying Lemnia 1,4(cl) to Eqs(3.9) and ( 3 . X ) ) . Xotv frorn

Eqs.(3.8)-(3.10) that P + Q = 1, is equivalent to PQ = QP = O arid r ( P ) + r ( Q ) = n t , . This ass~rtion

is also eqii ident to PQ = Q P = O and R(P) R(Q) = R ( P x ) cf;. R ( Q X ) = Cm'. which is Part ( c ) . Parts

( c i ) . (P.) and ( f ) follow from Eq.(3.8). O

As for the rmk of sum of tnro iclempotent matrices. wt-e have the following severiil resiilts.

Theorem 3.7. Let P. Q c PX be two idempotent mutrices. Then the smn P t (2 .snti.sfics the r(~nA.

cquulzties

P O P

Proof. Let 31 = . Then it is easy to see by block elerzier~triry oper;itioris of rriatriccs t h

Uri the other hand. note tliat P2 = P and Q' = Q. It is also easy to fincl b - block clrnientary optcttioris

The combination of the above three rank equalities yields the two equalities in Eq.(3.12). Conseqrier:tIy

applying Eq.(1.4) to the two block matrices in Eq.(3.12) yields Eqs(3.13) arid (3.14). respectil-ely. @

CoroUary 3.8. Let P. Q E C m x m be two idempotent matrices.

(a) If PC) = Q P. then

or, equiurïlently,

R(Q) c R ( P + Q) and R(Q') C R ( P œ + Q ' ) .

( b j If R(Q) R ( P ) o r R(Q') c R(P*) . then r( P + Q ) = r ( P ) -

Proof. If PQ = ()P. then Ecls.(3.13j and (3.14) reduce to

Co~nbining tliem with Eqs(3.4) and (3.7) yields Eq.(3.15). The equivdence of Ecp.(3.l5) arid (3.16)

follows from a siinple fact that

ZLS well as Lenima J..Z(c) and ( c i ) . The result in Part (b) follows irnmediately froni Eq.(3.12). 0

Corollary 3.9. Let P. Q E CnLXn' he truo idempotent rnatnccs. Then the follou~arq bu(: ~tute7r1cnt.s 1 1 1 . ~

(a) The s u m P + Q is nonsingular.

Proof. In Iight of Eq.(3.12), the sum P + Q is nonsingular if and only if

or equivalently

Ohserve that

Conibining them with Eqs.(3.11) and (3.18) yields the equivalence of Parts (a)-(e). O

Theorem 3.11. Let P. Q E CnL"'" De two idempotent mutrices. Then

(a) The r m k of I,,, + P - Q satisfies the equality

( I I ) The runk of 'ZI,,, - P - Q sutisfies the two eqnalzties

Proof. Replacing Q in Eq.(3.13) by the idempotent rnatris II,, - Q ancl applying Eq.(1.4) to i t >-ieitfs

cistablishing Eq.(3.19). Further, replacing P and Q in Eq.(3.12) by I,,, - P and I,,, - Q. lvc d s o 11'-

Eq.(l.4) firid that

- - rn - r ( Q ) + r ( Q - QPQ ).

establisl~ing Eq.(3.20). Sirnilarly, we can show Eq.(3.21). 0

Corollcary 3.11. Let P, Q E C n L X n L be two idempotent matrices.

(a) If R ( P ) C R(Q) und R ( P * ) C R(Q*), then P und Q satisfg the two rank equalities

( b ) I,, + P - Q as nonsimy~lar e r(Q PQ) = r(Q).

( c ) 21, - P - Q is nonsingular r( P - P Q P ) = r ( P ) r ( Q - QPQ ) = ~ ( d ) ) .

( c i ) Q - P = lm T(QPQ) + r(Q) = m.

Proof. The two conditions R ( P ) C R(Q) and R(PS) C R(Q') are eqiiivalent to QP = P = PO. 111

t h case. Eq.(3.19) rerluces to Eq.(3.22), Eqs.(3.20) (3.22) retiuce to Ec1.(3.'L3). TIic results in Parts

( a ) -(c j are direct consequences of Eq.(3.19). 0

we next consider the rank of PQ - QP for two idempotent matrices P and Q.

Theorem 3.12. Let P. Q E CnZXm be two idempotent matrices. Then the diflerence PQ - QP .suti.sfics

the f ive runk equ(~1ities

t.( PQ - QP ) = r( P - Q P ) +- r( QP - Q ) + r ( P Q ) + r(C)P) - r ( P ) - r ( Q ) .

Frlr particulnr. i f 150th P and Q are Hermitian idempotent, then

Proof. It is e,%y to verify that that PQ - QP = ( P - Q ) ( P + Q - I,,, ). Ttius the rank of PCJ - I JP

c m be espressed as

On the other hancl. it is easy to verify the Factorization

Herice

Putting it in Eq.(3.30) yields Eq.(3.24). Consequeatly putting Eq.(3.8) in Eq.(3-24) yields Eq-(3.25):

putting Eq.(3.1) in Eq.(3.25) yields Eq.(3.26); putting E q ~ ~ ( 3 . 2 ) and (3.3) respectively in Eq-(3.25) yields

Ecls.(3.27) and (3.25). 17

Corol lary 3.13. Let P. Q E Cr"Xr" be two idempotent matrices. Then tire following fiue state7ncnt.s arc

iq~riutrlent:

( a ) PQ = Q P .

(13) r ( P - Q ) + r ( I m - P - Q ) =m.

( c ) r( P - CJ ) = r ( P ) + r ( Q ) - r ( P Q ) - r (QP) .

( c l ) r( P - PQ ) = r ( P ) - r ( P Q ) and

( c ) r ( P - Q P ) = r i P ) - r ( Q P ) and r f

r ( Q - PQ ) = r (Q) - r ( P Q ) , ic.? PQ sr, P und P Q sr, Q.

r( Q - Q P ) = r ( Q ) - 7-(QP) , Le.. 4P Srs P and C)P Ir, (2.

and r [ P. QI = r ( P ) + r (Q) - r ( Q P j .

and r [ P. Q ] = r ( P ) + r ( Q ) - r ( P Q ) .

Proof. Follotvs imrnecliately frorn Eqs-(3.24)-(3.38). 17

Corol lary 3.14, Let P! Q E CmX" 6e two idempotent matrices. Then the following t h e statc~rrents

are equiualent:

(a) r( PQ - Q P ) = r ( P - Q ) .

( b ) I,,, - P - Q is nonsingular.

( c ) r ( P Q ) = r(C2P) = r ( P ) = r ( Q j .

Proof. The equivalence of Parts (a) and (b) folIows from Eq.(3.24). The ecluidence of Parts (1)) and

( c ) follows frorn Corollary 3.6(e). fJ

Coroi lary 3.15. Let P! Q E CnLXn' Iie two idempotent matrices. Then the following three stc~te7rlents

c m etpivalent:

( a ) PQ - Q P is nonsinyular.

(11) P - (2 c~nd In, - P - Q are rton~ingular.

( c ) R ( P ) 6 R(Q) = R ( P 8 ) 9 R(Q') = Crn crnd r ( P Q ) = r (QP) = r ( P ) = r ( Q ) l~old.

Proof. The equivalence of Parts ( a ) and (b) foflows from Eq.(3.24). The eqiiivalence of Parts (b) and

( c ) follows from Corollaries 3.4(e) and 3.6(e) . 13

-1 grotrp of analogous rank equdities cc= also be derived for P Q + QP. n-here P and & arc. tn-O

iclcrnpotent matrices P and Q.

Theorem 3.16. Let P, Q E Cn'X"' be two idempotent matrices. Tftert PQ + Q P sntisfies tfie rurtk

erpd2tie.s

I'( PQ + Q P ) = r( P t & ) i- r (P&) + r ( Q P ) - r ( P ) - r(Q).

T( PQ + QP) = r ( P - PQ - QP + Q P Q ) + r (PQ) + r (QP) - r ( P ) .

r( PQ + QP) = r ( Q - PQ - QP + P Q P ) + r (PQ) + r (QP) - r (Q).

Proof- Note that PQ + QP = ( P + Q )' - ( P +- Q ) - Then appiying Eq.(l . l l) to it . we directly obtain

Eq.(3.31). Coriseqi~entl~., putting Eq.(3.8) in Eq.(3.21) yields Eq.(3.32)? puttirig Eqs.(3.13) and 13-14)

rcspectively in (3.32) yields E q ~ ~ ( 3 . 3 3 ) and (3.34). a

Corollary 3.17. Let P, Q f CmX"' be two idempotent matrices. Then tire followiir~~ four .stcrte.rrrc7rt.s

(Lrc erpiuulent:

(a) r ( P Q + Q P ) = r ( P + Q ) .

(h ) Ir,, - P - (2 is nonsingular.

(c) r(PC2) = T(QP) = r ( P ) = r(Q).

(cl) r ( P Q - Q P ) = r ( P - Q ) .

Proof. The equivaIence of Parts (a) and (b) follows From Eq.(3.32). and the ecpivalence of Parts (1))-1~1}

cornes from Corollary 3.14. O

Corollary 3.18. Let P? 0 € C'"LXm be two idempotent matn'ces. Then tfle fo1lowi.r~~~ tiuo s t u t e r r ~ e ~ ~ t s (riri

q ~ ~ i u o l e ~ z t :

(a) PQ + Q P is nonsingular.

( b ) P + Q and Ir, - P - Q ure nonsingulur.

Proof. Follows directly from Eq.(3.31). 17

Combining the two rank equdities in Eqs.(3.24) and (3.3 1). wc ob tain the folluwing.

Corollary 3.19. Let P, Q E CnLX "' be two idempotent matrices. Then both of them satisf?, the folloruirry

rnnk idcntity

r ( P + & ) + r ( P Q - Q P ) = r ( P - Q ) + r ( P Q + Q P ) . (:3.35)

Thoerem 3.20. Let P. Q E CrnXRL 1)c two idempotent rnatrices. Then

~ [ ( p - q ) " - ( P - & ) ] = 7 . ( I m - P f Q ) f r ( P - & ) -7 lL.

r[ ( P - Q )'> - ( P - Q ) ] = r (PQP) - r ( P ) + r( P - Q ).

Proof. Eq.(3.36) is derived from Eq.(l.ll). -4ccording to Eq.(3.19). IW have r( I,,, - P + (2 ) =

1-(PQP) - r ( P ) + m. Putting it in Eq.(3.36) yields Eq.(3.37).

Corollary 3.21 (Hartwig c a l c l Styan, 1987). Let P, Q E C"'Xn' bc two idciripotent matrices. T ~ J L t h

following five stcrterrrents are equivalent:

(a) P - Q zs idempotent.

(1)) r ( I , - P + Q ) = m - r ( P - Q ) .

(c) r ( P - (2) = r ( P ) - T ( Q ) ~ 2.c: Q STs P. (ci) R(Q) R(P) und R(Q') Ç R(P') .

( c ) PQP = C).

Proof. The ecpivalence of Parts (a) and (b) follows immediately from Eq.(3.36). aricl the equivalence of

Parts ( c ) . (CI) ancl ( e ) is from Corollary 3.3(d). The equivalence of Parts ( a ) and ( e ) follows from a direct

rriatris cornputation. O

Iri Chapter 4. wc! shd1 d s o establish a rank formula for ( P - Q )" ( P - Q ) and consider tripotençy

of P - Q. where P. Q are two idernpoten~ matrices.

Theorem 3.22. Let P. Q E CmXn' he two idempotent matrices. Then ï,,, - PQ strtisj5e.s the n ~ n k

eqwtlities

r ( I , , - P Q ) =r(21,,, - P - Q ) = r [ ( I n , - P ) + ( I n , - Q ) ] . (3.3s)

Proof. Accordirig to Eq.(1.10) we have

r ( I m - P Q ) = r ( Q - Q P Q ) - r ( Q ) + r n .

Coriseqiiently piitting Eq.(3.30) in it yields Eq.(3.3S). 0

Corollary 3.23- Let P. Q E C'ln "' he two idempotent 7natricc.s. Then the surrr P + C) .sntisjir:.s thcl 7-unk

Proof. Replacing P and Q iri Eq.(3.38) bu two idempotent rriatrices I,, - P anci I,,,

yielcls Ec1.(3.39). If PQ = QP. then we know by E q ~ ~ ( 3 . 1 3 ) and (3.14) tiiat

tirid by Corollary 3.13 we ako know that

r( P - P Q ) = r ( P ) - r (PQ) and r ( Q - QP ) = r ( Q ) - r (Q P ) .

Piitting Eq.(3.42) in Eq.(3.41) yields Eq.(3.40). a

Corollary 3.24. Let P, Q E "' be two idenrpoter~t matrices. Then

In particular. the jollowing statements are equivaient:

( a ) PQ is idempotent.

( b ) r ( I,, - PQ ) = m - r (PQ) .

(c) T ( SIn, - P - Q ) = n~ - r (PQ).

Proof. Applying Eq.(l . l l) to PQ - (PQ)' gives the first equality in Eq.(3.43). The second one follows

frorri Eq.(3.38). O

Coroliary 3.25. Let P. Q E CnLXIn be two idempotent mutrices. Then

Proof. This follows from replacing -4 in Eq.(1.4) by In,. O

Yotice that if a matrix -4 is idempotent, then -4' is d s o idempotent. Thus n-e can easily find the

following.

Corollary 3.26. Let P E CnZXm be an idempotent mntrix. Then

(a) r ( P - P* ) = 2rf P. P* ] - 2r (P) .

(b) 1'(1,,, - P - P ' ) = r(I , , i- P - P ' ) = m.

(c) r ( P + P*) = r [ P , P*]. i-e., R(P)C_ R ( P + P X ) undR(Pa) C R ( P t P S ) .

(cl) r ( P P * - P'P ) = r ( P - P' ).

Proof. Parts (a) follows from Eq.(3.1). Part (b) fo11on.s from Ecl.(3.S). Par t (c) foilows frorn Eq.(3.31).

Part (d) follo~-s from Eq.(3.24) and Part (b).

The results in the preceding theorems and corollaries can easily be extendecl to matrices n-ich propertics

P2 = XP and Q' = pQ. where X # O and IL # O. In fact. observe t h r

Tlius both P/X ancl Q / p are idempotent. In that case. iipplying the results in the prcvious ttieorerris arid

corollaries. one may establish a variety of rank equalities and their conseqrrences related to siich liiriti of

matrices. For example.

and so on. We do not intend to present them in details.

Chapter 4

More on rank eaualities reiated to idempotent mat &ces and t heir applications

Tlic rank equalities in Chapter 3 can partially be estended to matris expressions thac irivolve idempotent

rrintrices and general matrices. In addition? they c m ais0 be applied to establisli rank equaiities relateci

to irrvolutory rriatrices. The corresponding results <are presentecl in this cliapter.

Theorem 4.1. Let -4 f C"'"" be yiuen. P E C"'" "' crnd CJ E C""" De two idenlpotcnt ~nrrt7tcc.s. Therr

the rliflercrrce P-4 - -4Q .sat.isfies the two rank equalities

1 P '44 O 1 rrixtrices that

Ou tlic othcr iiand, riote that P2 = P and Q2 = Q. It is also easy to firid by block elernentary opr.r;itioris

of rnatrices that

Corribining Eqs.(4.3) and (4.4) yields Eq.(4.l). Consequently applying Eqs.(l.2) m c l (1.3) to [.-IO. P J r 1

Corollary 4.2. Let -4 E Cnxn be giuen, P E Cmxm und Q E Cf' be two idernpotcnt matrices. Then

(a) R( P-4 - P-4Q ) n R( PAQ - .4Q ) = { O ) a n d R[( P.4 - P-4Q ) * ] n R[( P-4Q - -4Q )'] = (0) .

(il) If P-4C.j = O . then r( P.4 - -4Q ) = r (P-4) + r-(-4Q). or, equivalentl~l R(P-4) n R(,4Q) = ( O ) cr.rrcl

R(( P-A)*] n RE(-4Q)*] = ( O ) .

( c ) P.4 = -4Q a P,4( I - Q ) = O ctnd ( I - P).4Q = O w R(--lQ) 2 R ( P ) ( L T L ~ f?f(P--I-l)*] S

Ncs*).

Proof. Par t (a) follows £rom applying Lemma 1,4(d) to Eq.(4.2). Parts (b) and ( c ) are direct corise-

qiiences of Eq. (4.2). n

Corollary 4.3. Let -4, P, Q E C m X m 6e yiven with P, Q being two idempotent matrices. Then the

following three statements are equivalent:

(a) P-4 - -4Q is nonsingular. r 1

Proof. Follom-s from Eq.(4.1).

bscd on Corollary 4.2(c). we firid an interesting result on the generd solution of a rriatris cc~~iation.

Corollary 4.4. Let P E Cm*" and Q E CnXn Oe two idempotent matrices- Then the genernl .so17~tio7~

of the mrtr ix equation P S = SQ can be written in the two forms

X = PUQ + ( I n , - P)G'(I,,, - Q ) , (-1.5)

where U, 1.: tt' E C n L X n are arbitranj.

Proof. According to Corollary LL.'Z(c), the matri.; equation P S = SQ is equivalent to the pair of matris

cqirütions

P S ( 1 - Q ) = O and (1-P).YCZ=U. (4.7)

Solving the pair of equations. we cari find that both Eq.(4.5) and Eq.(4.G) arc the gcneral solutioris o f

P X = S Q . The process is somewhat tedious. Instead. wct give here clle verification. Puttirlg (4 .5) iii

P X and S Q . we get

P X = PUQ and SQ= PUQ.

TIius Ec1.(4.5) is solution of PX = SQ. On the other hand. suppose that -5, is a solution of P S = S G )

iitid let CT = I..' = .YO in Eq.(4.5). Then Eq.(4.5) becomes

n-liich iinplies tliat any solution of P S = -YQ can be espressed by Eq.(4-5). Hcnce Eq.(-l.S) is iritlced

the gcneral solution of the equatiori P X = SQ. Siniilarly Ive can verify that E,q.(4.G) is iilso a gerieral

solution to P X = S Q . O

-4s one of the basic linear matrix equation, -4S = S B wt.as esamined (see, e-g.. Hartwig [37]. Parker

[Z]. Slavova e t al [SS]). In general cases, the solution of --iS = S B cari only lie dctermiried b>* t he

canoriicai forms of -4 and B. The result in Corollcary 4.4 rnanifests that for idempotent matrices -4 and B.

tlie general solution of .AS = S B can directly be written in A and B. Obviously, the result in Corollary

4.4 is ülso vctlid for itn operator equation of the form -4S = S B when Loth -4 a11d B are idempotent

operators.

Theorem 4.5. Let -4 E PX be yiuen, P E CnLXm und Q E C n x n be t,wo idempotent matrices. T h r

tire surn P-4 + -4Q satisfies the rank equalities

Proof. Let M = [ Y " khen it is easy CO see by b~ocke~ementary opera:ions o f n~au-ices

P - i Q

On the other hand. note that P' = P and Q" = Q. We dso obtain by block elenientary operations of

riat tri ces that

Combining the above three rank equalities for ,il.! yieIds Eq.(4.8). Consequently applying Eqs.(2.2) and

( 1.3) to the two block matrices in Eq.(4.8) yields Eq44.9). 0

Coroilary 4.6. Let .i E CVkXn be given. P E C'nX"' und Q E C R '" t ~ e two irlenipotent nrutriccs.

(a) If P-4Q = 0. then r( P-4 + -4Q ) = r(P-4) + T ( - A Q ) , o r e p i v r d e 7 ~ t i ~ R(P-4) n R(.4Q) = (01 ( r d

R[(P.4>*] n R[(-IQ)'] = {O).

(b) P A t -4Q = O u P-4 = O and Ad) = 0.

( c ) The generul solution of the matriz equation PX + +YQ = O is S = ( 1 - P )Ci( l - Q ), where

U E CnkX is urbitrarg.

Proof. If P.4Q = O, then r( P-4 - -4Q) = r(P.4) + r(AQ) by Theorem .i.l(b)- Corisequently r( P-4 + -4Q ) = r(P-4) + r(,4Q) by Lemrna 1 .-l(d). The result in Part (b) foIlows from Eq.(4.9). -4ccorcfing to (11).

the equation P X + S Q = O is equivalent to the pair of matris ecluations P-Y = O and S C ) = O. Xccordirig

to Rao and Mitrü (1972), rtnd Mitra (1984), ttie comnion general solution of the pair of equation is esactly

-Y = ( 1 - P)L/*(I - Q ) , where Ci E PL"" is a r b i t r a - 13

CoroIhry 4.7. Let -4, P, Q E CniXrn De given with P. Q being two idempotent matrices. Then the

following five statements are equivalent:

(a) The sum P-A t -AC) is nonsingular.

Proof. Follows from Eq.(4.8). D

(a) P.4 -t -4Q = -4 ( 1 - P )A( I - Q ) = O and P-4Q = 0-

(b) The general solution of the mat& equation P-Y +- S Q = S is S = ( 1 - P )C'Q + I,'( I - Q ).

whc7-c u, Cr E Cr" '' m e u r f ~ i t r f ~ n j .

Proof. Accorcling to Eq.(4.1), we first nnd that

According to Eq.(1.2) and (1.3), we a h get

[(I-:)-~] = r [ â S ] - r(P), and ri.462, 1 - Pl = T(P--IQ) + r-( I - P ).

Conlbining the above three yields the first equality in Eq.(4.10). Consequently applying Eq.( l .4 ) to the

block xriatriu in it yields the second equality in Eq.(4.10). Part (a) is a direct consecpence of Ec1.(4.10).

Part (a) follows from Corollary 4.4. fl

If replacing P and Q in Theorem 4.5 by 1, - P and 1, - Q, we can d s o obtain two rank eqiialities

for 2-4 - P-4 - -4Q. For sirnplicity we omit them Liere.

Theorem 4.9. Let -4 E C m X n be gzven, P E Cm'"' und Q E LnXn be two idempotent matrices. Tlren

the runk of -4 - P=IQ satisfies the equality

In pnrtic.irlar.

(a) P - ~ Q = - ~ - I ( I - P ) - ~ ( I - Q ) = O . ( 1 - P ) - 4 Q = O and P-4( I - ( ) )=O-P. - l=

-4 ( 1 1 1 d -4Q = -4-

(b) The genercrl solution of the mat& equation P S Q = .Y is S = PUQ, where l.7 E is

arbitrnnj.

Proof. Note that P' = P and Q' = Q. It is easy to find that

as required for the first equafity in Eq.(4.11). Consequently applying Eq.(1.4) to its leit sicle yieltis t tw

seconcl one in Eq.(4.11).

-1pplying Eq.(4.1) to

results.

Theorem 4.10. Let P.

Part (a) is a direct consequence of Eq.(4.11). Part (b) can trivially be vwified.

powers of ciifference of two idenipoteh niatrices. we also firid follou-ing scverd

Q E C n L X m be two idempotent matrices. Then

( a ) ( P - Q ):' satisjïes the two rank equalities

171 pu~f icu lar~

(b) If (PQ)" PQ, then

( c ) r[ ( P - Q = r( P - Q ) ' Le.. Ind( P - Q ) 5 1. if and 07~11/ i f

or. eqîriualentl~j,

R ( P ) und R [ ( P - P Q P ) ' ] ~ R ( Q * ) .

. and R[Q - QPQ. Pl = R[Q. P l . (4 .16) (2 1

Proof. Since P' = P and Q2 = Q, it is easy to verify that

Lctcing -4 = In, - C,P arid applying Eqs.(4.1) ancl (4.2) to Ey.(4.l7) imniecliately yields Eq.(-l.l2) anci

(4.13). The resiilts in Parts (b)-(d) are natural consequences of Eq.(4.13). 13

Corollary 4.11. Let P, Q E CnLX"' be two idempotent matrices. T f ~ e n

Jn particulur.

(a) P - Q is tripotent R(QPQ) Ç R(P) and R[(PQP)'] R(Q').

(1)) If PQ = QP. then P - Q is tripotent.

Proof. Observe from Eq.(4.1/) tha t

Xpplying Eq.(4.1) to it irnmediately yields Eq.(4.18). The results in Parts (b) and (c) are riatirrd

consequences of Eq.(4.IS). O

Corollary 4.13. A mat* il E Cn'X"' is tripotent if and onlg if i t can factor as -4 = P - Q . whcrc P

( L T L I ~ Q m e two idempotent matrices witlz P Q = QP.

Proof. The .'if7 part cornes from Corollary .l.ll(b). The '- on1~- if" part folIows frorn a decunipasiciori

of -4

where P = +( -Aï + -4 ) and Q = :( - -4' - -4) are two idempotent matrices with PQ = Q P . 17

The rank equality (4.12) can be estended to the matris ( P-Q )y v-here both P and Q are idempotent.

En fact. it is easy to verify

Hence by Eq.(-L.l) it follows that

3lorcover, the above work can also be extendeci to ( P - Q )'"+'( k = 3. 4. - - -): wliere both P u i c t () are

idempotent.

-4pplying; Eq.(4.1) to P Q - QP: where both P and Q are idempotent. we also obtairi the following.

Corollary 4.13. Let P. Q E C n L X m be two idempotent rrrutrices. T h e n

r ( P Q - Q P ) = r ( P C ) - P Q P ) + r ( P Q P - Q P ) .

r ( P Q - Q P ) = r ( P Q - Q P Q ) + I . ( Q P C ) - Q P ) .

frr pwt.iculn7: i f hoth P rrnd Q rire Hermitian idempotent. then

Tlie r m k equdity (4.23) ras proved by Bérub6, Hartwig and Styan(1993).

Corollary 4.14. Let P, Q E Cm x m 6e two idempotent matrices. Then

7 - [ ( P - P & ) + X ( P Q - Q ) ] = r ( P - Q )

frolds for ull X E C with X # O . I n particular,

r ( P + Q - 2 P Q ) = r ( P + Q - 2 Q P ) = r ( P - Q ) .

Proof. Observe that

Thus it iollows by Eq.(4.1) that

Contrasting it with Eq,(3.1) yïefds Eq.(4.24). Setting X = -1 we have Eq.(4.25). 17

RcpIacing P by I , - P in Eq.(424), me also obtain the iollowing.

Corollary 4.15. Let P, Q C"'X'n be two idempotent mutrices. Thera

111 the rcmainder of this paper. we apply the results in Chapter 3 and ttiis chapter to establisli various

rank ec~uaiities related to involutory matrices. A niatris -4 is said to be involutory if its sqiiarc is idcntity.

i.c.. -4' = 1. -4s two specid types of matrices, involutory matrices and idempotent matrices are closely

linkcd. -4s a matter of hc t l for any irivolutory matrix -4. the two correspondirig matrices ( I + -4 j /2 aiid

( I - -4 ) / 2 are iclcrnpotent. Conversety. for any idempotent matrix .4. the t x o corrcsponclirig niütrictxs

+( i - 2-4) are irit~olutory. Based on the basic fact. al1 the results in Chapccr 3 nricl this c:liapter ori

idempotent matrices can dually be extended to involutory matrices. \Ve riest list sorrie of them.

Theorern 4.16. Let -4. B E C r n X n L be two involutanj rrrutrices. Thcn the m n k of -4 + B ~ 7 1 d -4 - LI

srrtisfy the epu l i t ies

r ( -4 - B ) = r [ ( I + A ) ( I - B ) ] + r [ ( i - . 4 ) ( 1 + B ) ] .

Proof. Notice that both P = ( 1 + ,-L)/2 and Q = ( I - B ) / 2 are idempotent when -4 and B are

involutory. III that case,

Piitting them in Eq.(3.1) produces Eq.(4.26). Furtherrnore we have

P~itt ing them in Eq.(3.2) yields Eq(4.27). moreover. if B is involutory, then -B is aIso involutory. Thus

replacing B by -B in Eqs.(4.26) and (4.27) yields Eqs.(4.'38) and (4.29). [7

Corollary 4.27. Let -4. B E C m x m 6e two inuolutonj matrices.

(a ) I f ( I + - 4 ) ( 1 - B ) = O o r ( [ - B ) ( I + , 4 ) = O . then

Proof. The condition ( I + -4 ) ( I - B ) = O is equivalent to

In that mse,

( I + - - L ) ( I + B ) =I+.A+B+.4B = 2 ( 1 + _ 4 ) ,

Thus Eq.(4.27) reduces to (4.30). Sirnilady we show that iirider ( I - B ) ( I i- -4 = O. the rank cqtiality

(4.30) also holds. The result in Part (b) is obtained by replacing B in Part(a) by -B. Ci

Corollary 4.18. Let -4 B E C"'X"L be two inuolutory matrices.

(a) Thc surn -4 + B is nonsingular if and onlg i f

(11) The clifference A - B is nonsingular if and onllj .if

Proof. Follows immediately from (4.26) and (4.27). 0

Theorem 4.19. Let -4, B E C m be two involutonj matrices. Then -4 + B and -4 - B sntisfy the runk

equalities

Proof- Putting P = ( 1 + -4 )/2 and Q = ( 1 - B ) /2 in Eq.(3.8) and simpiifying yields Eq.(4.36)-

Reptacing B by -B in Eq.(4.36) yields Eq.(4.37).

The combination of Eq.(4.26) with Eq.(4.36) produces the following rank equality

r [ ( f f B ) ( I + - A ) ] = r ( r f B ) + r ( I + . A ) - m + r [ ( I - . 4 ) ( I - B ) ] . (4.38)

Theorem 4.20. Let -4, B E C " ' X m Oe two involutonj nratrices. Thcn

Proof. Piitting P = ( 1 + -4 )/2 and Q = ( 1 - B ) /2 in Eq.(3.24) and si~nplifying yielcls Eq.(4.39). CI

Putting the given formulas in Eqs.(4.26)-(4.29): (4.36) and (4.37) iri Eq.(4.39) rriay yield soirie otiier

rad< eclualities for -4B - B-4. For simplicity. we do riot Iist chem.

Theorem 4.21. Let -4. B E C"'""' 6e two involutonj matrices. Then

1 -(,A-B) is i d e m p o t e n t e r ( I - A + B ) + T ( - 4 - B ) = m e r(.-l- B ) = r ( I t - 4 ) - r ( I + B ) . 2

Proof. Putting P = ( I + -4 )/2 and Q = ( I - B ) /2 in Eq.(3.36) and simpli-ing yields (4.41)-

Theorem 4.22. Let -4, B E C m x m 6e two involutonj matrices. Then

Proof. Putting P = ( I + ,4)/2 and Q = ( 1 - B )/2 in Eq.(3.38) and simplifying yields (4.43).

Theorem 4.23. Let -4 E CnLX"' be a n znvolutonj matrix. Thcn

(a) r( -4 - A= j = 2r[I + -4, I + A= ] - 2r( 1 -+ -4 ) = r[ I - -4, I - -4- ] - 2r( 1 - -4 1.

(11) r( -4 + -4- ) = rn-

(c) r( -4-4- - -4- -4 ) = r( -4 - -4- } -

Proof. Putting P = ( I +- -4 ) /2 and Q = ( I - B )/2 in Corollary 3.36 and simplifying yielcls the cIesirec1

results. O

Theorem 4.24. Let -4 E C l n x m and B E C'lx " 6e two znuolutoq matrices. .Y E Cm' " ' . Then .-Lay - -SB

srrtisfies the mnk equalities

Proof. Putting P = ( In, + -4 )/2 and Q = ( In + B ) /2 in Eqs.(4.l) and (4.2) yieIds Ecl.(-1.4-1) aricl (4.45).

The equivalerice in Eq.(4.46) follows from Eq.(4.45). CI

Theorem 4.25. Let -4 E C m X n l and B E C n X n be two ~ r i u o l u t o ~ ~ mutrices. Thcn the yene7-(11 so l l~ t io r~

of tlie matriz e q ~ ~ u t i o n -4S = ,YB is

S = 1.' + ,-l\/*B, (4.47)

Proof. ive only give the verification. Obviously the niatris ,Y in Eq.(4.47) satisfies .IS = .-II' + 1'B

and S B = 1,'B + _-IV Thus -Y is a solution of -4-Y = -YB. On tlie otlier hanci. for any solution .Yo of

-4-Y = -YB, let V = -&/2 in Eq.(4.47), then we get I/' = -4X0B = -Yo, tha t is. ,YU cari be represeritcd by

Eq.(4.47). Thus Eq.(4.47) is the general solution of the rnatrix equation -4-Y = SB. CI

Theorem 4.26. Let A .I Cntxn' Oe an inuolutonj matrïx, und S E CrTLX *'. T I L c ~

(a) .4S - -Y-4 satisfies the rank equalzties

In particdur.

(1)) The !jenerd .solutio~r of the mutrix equation -4-Y = .Y,4 is

Chapter 5

Rank equalities related to outer inverses of matrices

An oriter inverse of a matris -4 is the solution to the matrix equation S-4S = -Y. and is ofteri cleiiotecl

by -Y = -4('). The collection of a11 outer inverses of -4 is often denoted by -A( - ) . Obviously, the Sloore-

Penrose inverse. the D r a i n inverse, the group inverse. and the weighted 1Ioore-Penrose inverse of a

rr~atriu rue naturdly outer inverses of the niatris. If outer inverse of a rriatris is rilso Lin inrirbr irivcrsc

the niacris. ic is called a refiesive inner inverse of the rriatris. and is often tle~rutcd b!- -4;. Tlic. T h

collection of al1 reflexive inner inverses of a matrix -4 is denoted b j .4{1. 2 ) . -4s orle of important kirids of

generalized inverses of matrices, outer inverses of matrices and their applications have \\-el1 been c~sarriinccl

in the literature (see. e.g., [IO], [13], [30], [Sl], [76], [100], [102])- II: this chapter. WC shall establish sewral

tmsic rank equalities related to differences and sums of outer inverses of a rriatris. and then consicicr

tlieir various consequences. The results obtaincd in this cliapter will dso be appliccl in the subsequent

chap ters.

Theorem 5.1. Let .-l E C r r Z X n be giuen. and SI, & E =1{2). Then the difference -YL - - Y .s(~tz.sfies the

On the otlier hand, note tha t 4yl-451 = .YI and -Y2-4S2 = S2. Thus

which implies that

Cornbining Eqs.(5.4) and ( 5 .5 ) yields E q . ( ~ l ) . Consequently applying Eqs.(1.2) and (1.3) to t lie two

iilock matrices in Eq.(5.1) respectively and noticing that -4 f .Y1{2) and -4 E S2{S). n-e c m write

Eq.(5.1) <as Eqs.(5.2) and (5.3). a

It is obvious that if ,-1 = 1, in Theorem 5.1, then XI. -Y2 E 1,,,{2) are acciially tn-o itierripoter~t

niatrices. In that case. Ecp(5.1)-(5.3) reduce to the results in Theorern 3.1.

Proof. The resi~lts in Parts (a) and (b) follow immediately frorn applying Lemma 1.4(d) to Eqs-(5.2)

a d (5.3). Parts (c) is a direct consequence of Eqs(5.2) and (5.3). 17

Corollary 5.3. Let -4 E Cm"" be giuen, and Si. S2 E .4{2). Then the following five stcrtements (ire

(a) I - ( *Yl - 4Y2 ) = r(4Yl ) - r (-Y2): i e.. *Y2 srs -Yl -

L J

(c) R(,Y2) C R(-Yi) and R(S;) Ç R(X;) .

Proof. The ecpivdence of Parts ( a ) and (b) follows clirectly frorn Eq.(S. 1). The ecliiivrdence of Parts (b ) .

(c) and (cl) follows clirectly from Lemma 1.2(c) and (cl). Cornbining the two cc~iialities in Part (ci) yicllcls

ttie equality in Part (e). Conversely. suppose that Sl.4S2.-ISl = -Y2 holcls. Pre- and post-rnuItiplying

.Yl -4 and -4 -y~ to it yields Sl,4S2~4-1S1 = S1.4,Y2 = Sr-4-Yl. Combining it witti Si .-LS2--iSi = .Y2 J-iclds

the two rank equalities in Part (d). n

Corollary 5.4. Let -4 E CmXrn Oe giuen, and S I , ,Y2 E -4{2). Then the following tlrree stntemc7rt.s are

(a) The differcnce -Y1 - -Y2 is nonsingular.

Proof. -4 trivial consequence of Eq.(5.1). O

Corollary 5.5. Let A E Cm X n be given, and X E -+l{2). Then

Proof. It is easy to veri@ that both -4 and -4-Y-4 are outer inverses of -4t. Thus by Eq.(5.1) we obtain

the tlesired in Eq.(3.6). [3

Corollary 5.6. Let -4 E C"'""' be given, and .Y E .4(2). Then

Proof. It is easy to verify that both -4X and +Y-4 are idempotent LX-hen S E . A { ? ) . Tliiis wc firici by

Eqs.(3.1), (1.2) and (1.3) that

as recliiired for Eq.(ii.S). Eq.(5.9) is a direct conseqiience of E,q.(5.S). O

Corollary 5.7. Let -4 E C m X n be giuen, and S1, - Y E .4{2). Then

Proof. Notice that Both -LLYL-4 and .4&-4 are outer inverses of .4t when Si. Si E -- l {2) . Il<ireowr.

observe that r ( - U l A ) = r(,4,YI) = r ( S l - 4 ) = r(,Y1), and r(,4S2-4) = r(--l-Y-) = ~ ( ~ Y 2 - 4 ) = r(-i-z)- Thus

i t follon-s from Eq.(5.l) that

as reqiiired for Eq.(5.10). The verification of Eq.(S.ll) is crivial, tierice is ornitted. O

CorolIary 5.8. Let -4 E C m x n be yiuen, und &. .Y7 E .4(2}. Then the folfo~u.i.rcg five stclternents c l r r

c r p i ~ d c n t :

(a) r ( -4-YI -A - -4-y2 -4 ) = r ( - d y I -4) - r(-4-y2 -4). z- e.. --LY2 -4 Srs -Ady1 -4- F -

Proof. Foilotvs form Corollary 5.3 by rioticing t h t Both .4SL.4 and .-LY2.4 are auter irivcrses of -iT

wlien -1-1 . -Y2 f -4 ( 2 ) . [3

Tlieorem 5.9. Let -4 E C m X n be yiven, und -Yl, S2 E .4{2). Then the sum ,Yl + -Y2 srrtisfies tire rank

equulities

Proof. Let l'LI = O , . Then it is easy to see by blocl; elenientczry operations that

S1 Y O 1

which

Comb

implies that

r(it1) = r I ning Eqs.(5.

'Lx1 O

O O

s, .Y2

4) and ( 5 . 15) yields the first equa lity in Eq.(3.11). By symrnetry- we have riie second

equality in Eq.(5.15). Xpplying Eq.(1.3) to the two block matrices in Eq.(5.ll). respectively. ancl noticing

that -4 f {Sr} ancl -4 E {-K): we then can mi te Eq.(5.11) as Eqs.(5.12) and (5.13). Q

Corollary 5.10. Let -4 E C m x n 6e given, and Xl -Y2 E A(2).

(a) i f XL-4-Y2 = .Y2.iS1. then .(-Yi + -Y2) = r = r [SL. -Y2 1. -Y2 1 I '

Proof. Under SL14,Y2 = 4\-.>.4-Yl, we find from Eqs.(S.l2) and (5.13) that

Xote by Eq.(l.2) and (1.3) that

Tlius we have the results in Part (a). Part (b) follows irrimediately Froni Eq.(5.12). 0

Corollary 5.11. Let -4 € Cm"" he giuen, and Si, -Y2 E -4{2). Tiren the followiny jive stntements erre

equiuu lent:

(a) The su7n Si i- -Y2 zs nonsz'ngular-

(b) r = m and R n R = {O).

( c ) r[,Y2, SI ] = m and R [SI nR[-7] = ( O ) . .Y;

Proof. Follows immediately froni Eq. (5.1 1). C]

Corollary 5.12. Let -4 cz CTnXn be giuen. u n d -Y f --i{2). Then

Proof. Notice that Both A and .4,y2-4 are outer inverses of .4t when X E --L{2). Tlius Eq.(5.I6) fdlows

from Eq. (5 - 11) - O

Theorem 5.13. Let -4 E CnLXn be givent and Sr, -Y2 E ,-L{î). Then the diflerence Si - ,& strtisjies

the rank equnlities

Proof. Letting S = SI - - 2 anci applying Eq-(1.10) yields

which is Eq.(5.17). Sote that _4X1 and ,LY2 are idempotent. It turns out by Eq.(3.13) that

Putting it in Ec~.(5.17) yields Eq.(S.lS). 0

Proof. The equivalence of Parts (a) and (b) follows irnmediately from Eq.(S.li). T h ecluivaIence of

Parts ( c ) . (ci) <anci (e) is from Corollary 5.3. We next show the equivalence of Parts (a) and ( c ) . It is casy

to verify that

Tlius SI - -Y2 E .4{2) holds if and only if

Pre- and post-rnultiplying SI -4 and .4Y1 to it, we get

Piitting them in Eq.(5.19) yields Part (e). Conversely, if Part (e) holds. tlien Eq.(5.20} liolds. Conibining

Part (e) with Eq.(5.20) leads to Eq.(5.19}, which is equivalent to -Yl - -Y2 E .-1{'L}. 0

The problem considered in Corollary 5-14 could be regardecl as an estensioii of the work in CoroiIary

3.21, which was esamined by Getson and Hsuan (1988). In that monograph. they only gave a sirfficient

condition for ,YL - .:3 E -4(2) to hold when -YL, ,L:) E -4{2). Our result in Corollary 5.14 is a coniplete

conclusion on this problem.

Theorem 5.11. Let -4 f Cm"" be given. and S 1 , -Y2 E -A( - ) . Then the sum SI i -1-2 satisjies the two

rank equalities

Proof. Letting S = SI + -Y2 and applying Eq.(l.ZO) to S.4-Y - -Y vieIds Eq.(S.21). Note tliat --iS

aiid -4-y2 are idempotent. It turns out by Eq.(3.5) that

Putting it in Eq.(5.21) yields Eq.(S.22). 0

Proof. The equivalence of Parts (a) and (b) foliows immediately frorn cspariciing ( ,Yl + -Y2 ).A( Si + -Y2 ) - ( .VI + -Y2 ). The equivalence of (a) and ( c ) is from Eq. (5.2 1). ive nest show the equivalerice of ( b )

mt l (cl) - Prc- and post-rnultiplying ,YI -4 and - 4 S r to .Yl - 4 s + S-,,4-yL = O. we get

whicli implics t ha t -y1,4S-, = .Y2-4-YI. Putting t h e m in Part ( b ) yieltls ( c l ) . Coriwrscly. if Parc ( ( 1 ) iiolds.

then Part (b) naturally holds. CI

Chapter 6

Rank equalit ies relat ed t O a matrix and its Moore-Penrose inverses

In this chapter. ive shdl establish a variety of rank equdities related to a rna t rk arid its Moore-Penrose

inverse. and then use them to characterize various specified matrices. sucii as, E P rriatrices. coiijrigate

EP matrices, bi-EP matrices, star-dagger matrices, and so on.

As is well known, a rnatrùc -4 is said to be EP (or Range-Hermitian) if R(-4) = R(-4'). EP niatrices

have sonie nice properties. nieanwhile they are quite inclusive. Hermitian matrices and normal rriatrices

are special cases of EP matrices. EP matrices ancl tlieir applicritions have n-cll l ~ c ~ esarninrd in thcl

literature. Orie of the basic and nice properties related to a EP triatris -4 is -4-4' = .AT.-L. sw . e-g..

Ben-Israel and Greville (1980), C,mpbell and hIeyer (1991). This eqiialitj. niotivrtces LIS CO corisitlrr tiic

rarik of -4-4t - -4T-4. as well as its various extensions.

Theorem6.1. L e t - 4 E Cm""' be qiuen. T h e n the rank of ,44T --4t--l sntisf ies the foilawing rnnk erpalzty

Proof. Note that .-L-lt and -4t.4 are idempotent matrices. Theri applying Eq.(3.1). n-c first ohtain

Observe that r(_4_4t) = r(-4t-4) = T(-4). and

Thus Eq.(6.3) reduces to the first rank equality in Eq.(6.1). Consequently applying Ecl.(l.Z) to 1.4. -4' ]

i r r Eq.(6.1) yields the 0 t h - two rank equalities in Eq.(G.I). The equivalences in Part (a) are well-known

rcsults on a EP matris , mhich now is a direct consequence of Eq.(G.l). IL remains to show Part (b).

If r[=l-4t - -4t-41 = rn, then r[=l, -4' ] = r[-k-lt, -4t-41 = r[..l=lt - -4t-4, -4t.4 ] = m. Putting it in

Eq.(G.l), we obtain 27-(,4) = na. Conversely, if r [ A . -4'1 = 2r(.4) = m. then we irnmecliately have

r ( -4--It - At-4) = rn by Eq.(6.1). Hence the first equivalence in Part (b) is triie. The second equivalence

is obvious. Cl

Another group of rank equdities related to EP matriv is given below. tvtiich is rriotivaceci by it work

of Campbell and hleyer (1975).

Theorem 6.2. Let -4 E Cm ni be giuen. Then

(a) r[ -4-4ti -4 + = I ~ ) - ( -4 + .4t )ilAt 1 = 2r[ -4. -4- - %-(-A).

(1)) r[ -4t-4( -4 + -4t ) - ( -4 + .4t ) -4t-41 = 2r[A. .iœ ] - 27-(-4).

( : r[ -4-4t ( -4 t -4- ) - ( -4 + -4- ).i.4t ] = 2r[ A. -4- ] - Zr(--i).

(cl) r [ -4' -a( -4 t -4' ) - ( -4 + A* )-4t.4 ] = 2r[ -4: -4* ] - %(-A).

(e) The followiny statenlents are equiuulent(Carnpbel1 and Meyer. 1975):

(1) -4 2.5 EP.

(2) A-4q -4 -+ -4t ) = ( -4 4- .4t ) U t -

(3) ALI( -4 -+ .4f ) = ( --1 4- -4t )-4t-4.

(4) -4-4q -4 + -4- ) = ( -4 + -4- )-4*4t.

( 5 ) - 4 t 4 -4 + -4- ) = ( -4 + -i* )A+ -4-

Proof. FoIIows from Eq.(4.1) by noting that -4.4t and .-lt.4 are idempotent matrices. 0

Replacing the matrices ( -4 + -4t ) ancl ( -4 + -4' ) by ( -4 - .-lt ) and ( -4 - -4' ) in Theoreni 6.2. t h

rcsults are also crue.

In an earlier paper by Meyer (1970) m d a recent paper by Hartwig and Katz j 1997). tliey cstaldis1ic.d

riecessary r m d siifficient condition for a block triangular matris to be EP. Ttieir ii-ork non- cari btb

estended to the following general settings.

Corollary 6.3. Lct -4 € C"L"r'i. B € Cnixk . and D E Ch' k given. und let :\f =

(b) If R(B) c R(-4) und R(B') C R(D*), then

(c) (Meyer, 1970. Kartwig arid Katz, 1997) 11.1 is EP i f and only ij-4 and D cire EP. und R(B) C R(.-I) r -l

4 O ( r d R(B*) C R ( D * ) . In that case, Mrbft = I -

Proof. Follows immediately from Theorem 6.1 by putting .M in it. !3

Corollary 6.4. Let

he given. T h e n LW zs EP if and only if -A1l, -422 - - - , -Ann ure EP: und

-4 parallel concept to EP matrices is so-called conjugate EP macrices. X rriatris -4 is said ro tw corijil-

girte EP if R(-4) = R ( - - L ~ ) . If matrices considered are real. the11 EP matrices and coqjugate EP triatriccs

are identicai- hlucli similar to EP matrices. conjugate EP macrices also have some riice properties. one

of the basic and nice properties related to a conjugüte EP m a t r ~ x -4 is -4,-lt = -4t.4 (see t h series il-ork

[58]. [XI]. [GO]. (611. and [62] by hleenakshi and Indira). This equality motivates us to find t h following

resuits.

Theorem 6.5. Let -4 f C m x m be given- Then

- r( - 4 ~ ' - dt.4 ) = 2r[ A. .-lT ] - Zr(.-L).

( a ) .L4f = -4t.4 a r [ -4. ;1"] = T(-4) o R(.4) = ~ ( . - l ~ ' ) . Le.. .-i is curtjugate EP. -

( I I ) .4.-li - ;Li-4 is rionsingulor r[.4. .AT ] = 2r(.4) = rrr a R(.-L) t+ R(-4') = Cr'' -

Proof. Since ~ * . 4 and -4'-4 are idempotent, applying Eq.(3.1) to .4.lt - -4t.4. ive obtnin

mhich is esactly Ec1.(6.4). The results in Parts (a) mcl (b) folIow irnmediately from Eq.(G-4). 0

-4 i3 Corollary 6.6. Let A f CrnXm, B E cmxk. and D E C I i X k be given. and Jenote AI =

in pm-ticulur. !CI is conj~lgate EP if and only if A and D are con-EP. and R(B) C R(.4) and R ( B - ~ ) C

R(D').

Proof. Follows from Theorem 6.3 by putting LW in it. @

The work in Theorem 6.1 c m be e.xtended to matris exqxessions that involve the power of a matris.

Theorem 6.7. Let -4 E be given and k be an integer with k 2 2. Then

(b) ,4",-lt - ,4t-4"s nonsingdar r = r[=1". -4'1 = 'Lr(--l) = r n u r(-4") = r(.-l) n71d

Proof. Writing ,lk.4t - -di-4" = -[(.4~.4)..1k-L - .4"-1(--l..lf)] and applying Eq.(.l.l) to it. Xe obtain

as rcquired for Ec1.(6.5). The results in Parts (a) and (b) follo~v irnrnecliateIy froni Ecl.(G.5). C

Froni the result in Theorem 6.T(a) we can estend the concept of EP matris to pomer case: A square

rriatrix -4 is said to be power-EP if R(=Lk) 5 R(A') and Ri(-A")'] C R(.-l). where k 2 2. Tt is belieueci

t h t power-EP matrices. as a special type of matrices. might also have some inceresting properties. But

wr tio riot intend to cliscuss power-EP matrices and the related topics in tliis tlicsis.

Theorem 6.8. Let -4 E Cn'X"' be given and k be an integer with A: > 2. Then

-4" (b ) .4(Ak)t = (~") t . - l r = TV\ .?-A" ] = r(-4" ) R(-4") = R(-4'-4') and R[(.4")'] = I ;i*AU I

Proof. It foilows by Eq.(2.2) and block eIementary operations that

establishing Part (a).Part (b) follows immediately from Part (a) . Conibiriing Theoreni G.Ï (n) and G.S(b)

yielcis the implication in Part (c). CI

A scliiare matris -4 is said to be bi-EP, if -4 <and its Moore-Penrose inverse .-If satisfy (_4-4t)(.4t.4) =

(.4ta4)(-4.-lt). This specid type of matrices were examined by Campbell and Meyer (1975). Hartwig and

Spindelbock (1983)- .Just as for EP matrices and conjugate EP matrices. bi-EP matrices cari also be

characterized by a rank equalit-

Theorem 6.9. Let -4 E C V L X n z be giuen. Then

In ~ I L T S ~ C U Z ~ L ~ , the followzng four statements are equiualent:

(a) (-4-4t) (-4t-4) = (-4t-4) (-4-4t) i. e- . -4 zs hi-EP.

( b ) ( - 4 y € { (A2) - } -

( c ) r [ -4. .4* ] = 'Lr(,4) - r(,4').

(a) ciim[R(-4) n R(-Ac)] = ~ ( ~ 4 ~ ) -

Proof. Note that both -;l--lt and -4t.4 are Hermitian idempotent and R(-4t) = R(,4'). 1 1 - c b Iinve by

Eq.(3.29) t h a t

establishing Eq.(G.G). Xpplying Eq.(SS) and then the rank cancellation laws (1 .S) to -4- - -4- (.4t)'.-t2.

~ v e then obt,zin

as recluired for Ec1.(6.7). The equivalence of (a)-(c) follo~vs frorri Eqs.(G.G) and (6.7). The celui\-alcricc uf

(c) and (d) foliows from a well-known rank equality rf -4, BI = r(.4) + r ( B ) - clim[ R(-4) f l R ( B ) 1. @

The above work can also be estended to the conjugate case.

Theorem 6.10. Let -4 E C m x m be given. Then

Proof. Follows from Eq.(3.29) by rioticing that both -Ut and At-4 arc iclempoterit. Cl.

A parallei concept to bi-EP matris non- can be introduced: X square matris -4 is said to be cor~j4tgute

hi-EP if (-4--lt) (-4t-4) = (-4t-4) (-4-4t). The properties and applications of tliis special type o f niacric-cs

rcrriain to further s tudy

We ncst consider rank equalities related to star-dagger matrices. -4 sqiiarc rriatris -4 is said CO t>t>

star-dagger if -4*=lt = = ~ ~ = l ' . This special types of matrices were proposed ancl esarniried by Hartwig ancl

Spindelbock (1953), Iater by Meenakshi and Rajian (198s).

( c l ) Cn particular, the followin(~ stutement are equiualent (Hurtwig and Spindelbück-, 1983):

(1) .4'-4+ = ~ ~ - 4 * , 2.e.. -4 is star-dagger.

( 2 ) -4-4?4f-4 = A-4t-4*-4.

(3) -4-4"-42 = - 4 w - 4 .

Proof. We find by Eq.(2.2) that

as recluired for Part (a). Siniilarly me c m show Part (b). The result in Part ( c ) folloivs imrnccliatcly froni

Part (a). and Part (cf) foLIows from Par ts (a) and (b). 17

-4s pointed out by Hartwig and Spindelbock, 1953, star-dagger niatrices are quitc inclusive. Sormal

matrices. partial isornetries, idempotent rnatrices, 2-nilpotent matrices. and so on are crll special c a e s of

star-dagger matrices. this assertion can be seen from the statement (3) in Theoreni 6.11(d).

The results iri Theorem 6.11 can be extended to general cases. Belon- are tliree of them. The proofs

are rriuch s imi lx to that of Theorem 6.11, are therefore omitted.

Theorem 6.14. Let -4 E Cr"X"' be given und k be c m integer with k 2 2. Thcn

(a) r[.-LW(.4")t - (.4P)t.4-] = r [ A " ' ( ~ " ) ' r l " ~ ~ - rlP+' ( r l " )* . i~] .

(b) *4=(Ak)+ = (--Ik)tA= / tk(Ak)*Ak+l = -4k+L(Ak)*-4".

Sest are several results on ranks of mat r i - espressions involving poivers of the lloore-Penrose inverse

of a rnatrk.

Proof. By Ec1.(2.1) we easily obtain

Both of the above are esactly Part (a). Nest applying Eq.(l.lL) to In, - (-qt)'> ive obtain

establishing Part (b). The resii1ts in Parts (c)-(f) follow directly from Parts (a) aiici (b) . 0

Proof. It follows first from Eq.(l . l l ) that

r[.4t - ( ~ t ) ? ] = T( Im - =lt ) i- r(-4) - m.

4.4t + (-4t)'> 1 = r ( Im i ;lt ) + 1-(-4) - rti.

Then n-c have the first two eqiialities in Part (a) bu Theorerri 6.15(a). Sotct tliar.

.4[.4+ L- (.4')']-4 = -4 r - ~ ( A ~ ) ' A and * (.4t)'.4].4t = --Lt k (-4')'

Tlitis ive have the second equality in Part (a). Parts (b)-(g) fotlotv frorn P'vt (a). 0

Theorem 6.17. Let -4 E Cm"" Oe yiven. Then

(a) r[--It - (.At)" = r(-4' -I- -4.4*.4) + r ( .AZ - II-4-A) - r(-4).

( 1 1 ) r[ -4t - (-4t)3 ] = r ( -4 + -4;li ) + r( -4 - ;LA* ) - 4-41. if -4 is EP.

(c) r [ , . ~ ~ - (-4t)3] = r ( -4 +-4') +r (=I - -4') -r(-4) = T(-4- =L3) . ij.4 is Herrr~itian.

(cl) ( - A ~ ) ' = -4t - T-( .A2 + -4-4œa4 ) + T { A2 - - 4 X -4 ) = r(-4) R( -4-4* -4+A2 ) n R( -4Aœ-4 - -4' ) =

(0 ) and R[(-4.4'-4 + -4' ) * ] n RI(-4-4*.4 - -4' )'] = { O ) .

Proof. Applying Anderson and Styans' rank equality (1.14)

Tlien putting Theorern 6.16(a) in it yields Part (a). The resiilts in Parts (b)-(cl) fotlutv d l frorri Part

(a). D

Theorem 6.18. Let -4 E C n l X be giuen. T f ~ e n

(a) r( -At - .4* ) = 7-( -4 - -4.4- -4 ) .

(b) r ( -4 - -4- -4-4* ) = r ( -4 - -4--la -4A* -4 ) . l n purticulur.

( c ) -4t = -4' w -4A*.4 = -4, i-e., -4 is partial i s o m e t q .

(cl) -4t = -4- -4-4- e -4-4=-4-4- -4 = -4.

Proof. Follows froni Eq.(2.1). I7

Proof. Note that -4, -4t E =1{2) when -4 is idempotent. Thus we have by Eq.(5.l) tliat

establishing Part (a). Parts (b) and (c) foIlow from Eqs(6.6) and (6.7). Part (ci) is a direct consequencc

of Parts (a) - (c) . C7

Theorem 6.20. Let .4 E CmX"' be tripotent, that is, ..13 = -4. Tften

Proof. Xote that -4. --lt E =1{2) d i e n -4 is tripotent. Thus ive have by Eq.(.j.l) t l iar

cstablishing Part (a). Parts (b)-(e) follow from Eqs.(6.5), (6.6): (6.7) and Tlieorern 6.8(a). Part ( f i )

follows frorn Thcorcm 6.13(c). Part (f) is a direct consequence of Parts (a)-(e). 0

Tlic following result is motivatecl by a problem of Rao and Mitra (1911) or1 thc riorisingiilririty of :i

rriatris of the forni I + -4 - -4i.4.

Theorem 6.21. Let -4 f CnLXn' and 1 # X E C be yiuen. Then

(a) r( In, + -4 - -At-4 ) = T ( 1, + -4 - -4-4t ) = T(-4") - r(.-L) + m.

(1)) r( - -4 - --lt -4 ) = r( lm - -4 - -4-4t ) = r(-4?) - r(-4) +- m.

(c) 1.(XIni +-4--Jt .4) = I . ( , \ I ~ ~ +.A--4.4') = r [ ( h - 1)1,,1 +;LI.

(b) ( Rao and Uitra. 1971) T, i A - .-lt-4 i s nonsingular a 1, i -4 - .4.4* is r~onsing.ular

1-(-4?) = r(-4)-

Proof. XppIying Eq.(2.l) and then Eq.(l.S), we find that

By syrnmetrq-, we also get r( I,,, + -4 - -4-4t ) = m + r(..12) - r(-4). Both of thcm rire the resdt iri Part

(a)- SirniIarly Ive can show Part (c) by Eq.(2.1) and (1.11). Part (ci) is a direct consequence of Part(a).

Chapter 7

Rank eaualit ies related to mat rices and t heir ~oore-Penrose inverses

\ve consider in this chapter ranks of various matrix expressions that irivolvc two or more rriatrices ziricl

tlieir hloore-Penrose inverses. and present their various consequences. wtiich can reveal a series of iritririsic

properties related to Moore-Penrose inverses of matrices. Most of the results obtainecl in this chapccr are

new and are not considered before.

(b) --L-lt~ - B-4t-4 is nonsingular

r [ ] = r[A7 B-4*] = ?r(-4) = rn R(-4) + R(B-4") = Cm' and R(,-lB8) = R(A) R(-4*) fy R(B*,il) = C n L and R(.-I*B) = !?(.A').

Proof. Note that .-l.-lt and .4t-4 are idempotent and R(-4t) = R(-4'). Ive have by Eq.(4.l) tliat

cstablishing Eq.(Ï.Z). Parts (a) and (b) follow from it. 0

Clearly the resiilts in Theorems 6.1 and 6.7 are special cases of the above theorerri.

Theorem 7.2. Let -4 E C m X n , B E Cmx%nd C E ClX he given. Then

(b) r( -At--1 - CTC ) = 2r / c 1 ~ ( - 4 ) -T-(C).

(e) -4-4t = BBt e R(-4) = R(B).

( f ) -4t-4 = c ~ C R(.4') = R(C*).

(g) r(.-Irlt - B B ~ ) = r(--L4f) - r (BBt) o R(B) C R(.4).

( i ) r ( d -4 t - L?Bt ) = TTL o r[-4, BI = 4.4) + r (B) = m o R(-4) 6 R(B) = Cr".

(j) r ( --Ii-4 - CtC' = TL - r [ ] = r(-4) + r (C) = n - ii!(-4-) 62 îi!(C*) = C r i .

Proof. Xote that -4.4i. =L~=I . BBt. and CtC are ali idempotent. Thus we cari easily clcrive by Ecls.(3.11

m d (3.12) the four rank equalities in Parts (a)-(d). The resuIts in Parts (e)-(j) arc direct consequerices

of Parts (a) mcl (b) . Cl

Theorem 7.3. Let -4 f C m X n , B E CkXnL be given. Then

( . ~ = L ~ ) ( B ~ B ) = (B'B)(-4-4') Ç=, 4-4. B ' ] = r(;l) + r (B) - r(B.4) E=, dim[R(;l) n R ( B = ) ] = r(B-4) .

( 7.3 1

Proof. Xote thric -4.4'. -4t.4, BB'. and B t B are Herrnitian iclenipotent. Thiis ive tirid by Eq.(3.29) tliat

= 2r[.4. B' ] t 'Zr( B-4) - 'Lr(-4-1) - 27-( B).

as required for Eq.(7.2). The results in Eq.(7.3) is a direct consecluence of Eq.(72). 13

Theorem 7.4. Let -4 E CrrLXn, B E C k X m be g2uen. Then

(a) r[.4-( .-LA- i BB* )tA - .-lt-41 = r ( d ) + +(B) - r[-4, BI. r 7

( e ) .-lS(;L.4= + BB* ) tA = ' ~ ~ - 4 -4*(AA* + BB' ) tB = O rt.4. B ] = Y(-4) + r ( B ) u R(.4) n R(B) = {O).

( f ) -4(.4*-4 + C * c ) t = ~ * = ilAt .A( -4'-4 + C * C ) i C * = O a r 1 C 1 = r(-4, + r i c i - 1 J

R(.4') ri R(Cœ) = {O).

Proof. P a n s (a) and (b) are derived from Eq.(2.2). Part(c) and (d) are deriveci from Eq.(?.l). The

resiilts in Parts (e) and ( f ) are direct consequences of Parts (a)-(d). 0

Theorem 7.5. Let A E CrnXn. B E c m x b n d C E L L X n be given. Thcn

( b ) BBt-4 - -4CtC is nonsingular u r 1 1 = ,-[-AC*. B I = T(B) + i ( ~ . ) = i ~ i .

Proof. Follows from Eq.(4.1) by noticing that both B B ~ and CtC are idempotent. 0

It is well known that the rnatriv equation B-YC = .-i is solvabte if and only if BBr.-LCtC = -4 (sce.

cg., Rao ancl h8Iitra. l g i l ) . This lead us to consider the r m k of -4 - BBt.4CtC.

Theorem 7.6. Let -4 E C m X n , B c m X k and C E clXn be giuen. Then

BB'.~C'C = -4 o BB'A + -4CtC = 2.4 u R(A) C R(B) and R(;La) C R(Cœ) .

Proof. Applying Eq.(S.S) and the rank cancellation law (1-8) to -4 - BBt-4CtC protluces

r( -4 - B B + - - ~ C ~ C )

B',-LC' B* BB' O

n C C ] -r(BI - I ( C )

O BEI* - -4

as required for Eq.(?.5). In the same way we can show Eq.(7.6). The result in Eq.(7-7) is weI1 known.

R

Theorem 7.7. Let -A E C"lXn, B E C m x k and C E CL X n De gaven, and let AI = [: u] Tlmt

t h t is. the block matriz LW .sutisfies the rank equulity

Proof. Applying Eqs-(2.2) and (1.8) to -4 - BBt-4 - -4CtC yields

T ( A - B B ~ A - -4c t c )

B'BB* O

= [ 0 C'CC*

BB' .4CS

B'B O B--4

= r [ 0 Cc* C

B Ac- -4

B*-4

C C ] - r(BJ - r ( < )

-4

- 7-(B) - r ( C )

as recliiired for Eq. (7.8). 0

Theorem 7.8. Let -4 E C m X n , B cnax" and C E C' be giuen, und let 111 =

9-1.4 - A ( E B A F ~ ) ~ A ] = r(-4) + r (B) + r(C) - r(hf)7 (7.10)

thut 2s. the block mut% iCI satisfies the r a d equulity

luhere Eo = I - BBt and Fc = I - CtC. In particular.

r ( M ) = r(A) + r (B ) + r (C) , i-e., R(-4) n R(B) = {O) and R(_5IL) n R(C') = { O ) . ( X 3 )

Proof. Let !V = Es,4Fc. Then it is easily to verify that iV*!VNc = Iri that crise. applying

Eqs.(.2.1): and then (1.2) and (1.3) to -4 - = I ( E s ~ ~ c ) t = l yields

:w reqiiired for Eq.(7.10). The equivalence of Eqs.(ï.l2) and (7.13) folIows inimediately frorri Eq.(l. 11).

It is known that for any B and C , the matriv (Es-4Fc)t is always an outer inverse of -4 (Grcville.

1974). Thus the rank formula (7.11) can d s o be deriveci from Eq.(5.G).

Iri the remainder of this chapter, we establish various rank equalities related to rariks of lloore-Penrose

inverses of blocli matrices, and then present their consequetices.

TIieorem 7.9. Let -4 f C m x n , B E C m x k and C E C l X n be given. Then

Proof. Let Ad = [ A , B ]. Then it foiiows by Eq.(2.7) that

O O B* BB*

L J

= r-[.4-4u.il- Jdi\l'=I. BB'B- &li\.I*B] =r[-4-4-B. BB'.4].

as requirecl in Part (a). Similarly, we can show Parts (b), ( c ) and (ci). The resirIts in Parts (e) nrid ( f )

follow immediately from Parts (a)-(d). O

A gcneral rcsiilt is given below, the proof is omitted.

Theorem 7.10. Let .-i = [.-II, .4-, .. - . -Ak] E CnLX" be gieren, und denote hf =

Theorem 7.11. Let -4 E C n a x n l B E C m x k and C C C L X n be given. Then

Proof. Let LU = [ -4, B 1. Then it fotlows by Eqs-(2.7). (1.8) and biock eienien~ary operations of niatrices

r -4*,4 -4*B 4 4 O

= I -4 B -4 B O 1 - r ( M ) - r(-4) - r ( B )

4 B'B O B'B Ba

r 0 O O O - 4 - 1

required in Part (a). Ln the same way, we can show Part (b). 1% know from Part (a) t

iIJbft = -4.4' + B B I o r[ A. B ] = r(-4) + r ( B ) - Zr(;L* B ) .

liat

(7.16)

On the other hand. observe from Eq.(l.2) tliat

T h s Eq.(7.16) is also equivalent to -4-B = O. In the similar m~mner. n-e can show Part (ci). a

-4 general result is given beiow, the proof is omitted.

CorolIary 7.12. Let -4 = [ -41, .4? , - - - .Ak ] E C m * be yiuerr. Then

Theorem 7.13. Let -4 E C I n x n . B E C ~ I X A - and C E clX" 6e given. Then

Proof. Let i\f = [ -4, 8 1. Shen i t follows by Eqs.(2.7) anci (1.8) ancl block eierricritary operations of

matrices that

[ w - L * ] [U]

~1.r * -4(EB -4) ' Jf' B(EA B)' :If *

(EB-A)*-4(EB-4)' O (EB-4)' - r ( ~ 5 ~ - 4 ) - ï-(E.-i B )

O ( E B ) * B ( (EZ1 B)' 1 - - r

as reclrrired for Part (a). Similarly. we can show Part (b). Thc rcsults in Parts ( c ) ancl ( (1) follon.

ininiccliately from Parts (a) and (b). 0

- 0 !\f*.4(EB-4)- M S B ( E A B)' i\f ‘

O (E~.-l)*-4(f i~--l)* O ( E B - ~ ) *

O O (E ,~B)*B(EAB)* (EAB)'

-4 - O O O

B ' O O O -

A general results is given below.

Theorem 7.14. Let -4 = [.A1 .A2, - - - Y -Ak ] E Cm X n be giuen. Then

Theorem 7.15. Let -4 E CmXn. B E Cn'X%nd C E CL be giuen. Tiren

59

The proof of Theorem 7.15 is much sirnilar to that of Tlieorem 7.L3. hence ornittecl.

Theorem 7.16. Let -4 E C n L X n t B C ~ L X ~ and C E CL"" lie giuen. Then

Proof. Let M = [-4, B ] and IV = 1 -: . Tiieii we find by Thçorerri i ï ( b ) i h r

as reqiiired for Part (aj. Similclrly we can show Part (b). Parts (c) and (ci) arc direcc consccpiericcs of

Parts (a) and (b).

-4 generd resuit is given below, the proof is omitted.

Corohry 7.17. Let -4 = [-4L1 -A2 - - - -Ak ] E C m x n be given. Then

Theorem 7.18. Let , 4 E C m X n . B E cnLXk and C f C t X n be given. and let i\d =

r ( M ) = r(.4) t r ( B ) + r (C) . i.e.. R(.-l) fi R ( B ) = {O) , m t l R(.-L*) n R ( C W ) = { O ) . i, 7 .26)

Proof. It follows by Eq.(2.1) that

as required for Eq.(ï.23). 0

Theorem 7.19. Let -4 E CnLXn, B E Lm x%and C E C' Xn be giuen. Then

-4 B Proof. Let i\f = . Then i t follows by Eq.(3.1) arici block elernentar?. cqwration that

1 [O. B] M* - -4

as rcquirri for Eq. (7.27). CJ

It is easy to derive from Eq.(l.G) that

'I'ow replacirlg .4 by -4 - BXC in the above inequdity, where -Y is arbitrary, w e obtxin

This r , uk inequality implies that the quantity in the right-hand side of Eq.( i29) is a iouler boririd for

the rank of -4 - BSC with respect to the choice of S. Combining Eqs.(7.27) cznd (7.29). wc irnrnediately

obtain

rninr(.4 - BSC) = r [ A , B ] + r .Y [;] -. [; ;].

anci a matris satisfying Eq.(ï.30) is given by

s = - [ O ,

Theorem 7.20. Let -4 E C r n X " ,

[i :]' [fi] -

and C E C l X n be giuen. Then

and iV = [ *: ] . ï'iirii i r h i d by Ihrorern 72(b) t i r r

as rccpircd for Part (a). Sirnilady me c a l show Part (b). Observe ttiat

Tlius (c) and (cl) follow. 0

--\ generül result is given beiow, the proof is omitted.

Theorem 7.21. Let

bc! y2ve.n. m d let -4 = diag( .Al 1, .-Iz2, - - - . .4kk ). Then

(a) r [ M M - c i ( A , - - - . =L~*;L[, ) ] = 27-1 LU. -4 1 - r ( 8 f 1 - r(-4)-

Chapter 8

Reverse order laws for Moore-Penrose inverses of products of matrices

Reverse orcler laws for generalized inverses of products of matrices have been ari attractive topic in

the tlieory of gcrieralized inverses of matrices. for these Iaws can revcül essential rclationships bet~weri

p,ericralizecl irivcrses of protlucts of matrices and generalizeci inverses of eacli niatris in the prutiiicrs.

Various results ori reverse order Iaws related to inner inverses. refle-xive inner inverses. Moore-Perirose

irivcrses, group inverses. Drazin inverses. and weighted Moore-Penrose iriverses of products of rriatrices

have n-idely been established by lot of authors (see. c g . . ET], [SI. [W]. [?il. [?SI. [31]. [32]. [ A l ] . [BG].

[ S t ] . [go], Pa]. [94]. [102], 11031, [104]). In this chapter. we shdl present some rruik eclualities rclated to

~~rocliicts of Moore-Perirose inverses of matrices. and theri derive frorn them various types of reverse orcler

l a w for Moore-Pcrirose inverses of products of matrices.

Theorem 8.1. Let -4 E Cr" " and B E Cn X P be giuen. Then

in p~r- t ic~~lar . the following Eue statements ure equiuulent:

(a) Bt--It E {(-AB)-), i.e.. Bt.-lt is an inner inverse of -AB.

(1,) r[ .-l=. B ] = 7-(-4) + r ( B ) - r(-AB).

(c) clim[R(-4) n R(B')j = r(.4B).

( c l ) r( B - --lt - 4 ~ ) = r (B ) - r(-4tA4B), ï-e., -4t-4E3 srs B.

(e) r ( -4 - -WBt ) = r(-4) - r(ABBt), i.e., d ~ ~ t sr, -4.

Proof. Xpplying Eqs.(2.8) and (1.7) to ,4B - - 4 ~ ~ t . - l t - 4 ~ , n-e obtüin

- B'.-iœ B'B

- '1 -4-4- 4l3 ] + r(-4B) - r(-4) - r ( B)

Thus we have Eq.(S.l). The equivalence of Parts (a) and (b) follonrs irnmediately from Ecl.(S. 1). The

ecluivalence of Parts (b) and (c ) follows from the simple fact

The ccluivalence of Parts (b): ( c i ) and (e) foilows from Eqs(l.2) and (1.3)- @

The rank formula (8.1) was established by Baksalary and Styün (1993. pp. 2 ) in an alternative forni

Observe that

Thus Ec1.(8.1') is esactly Eq.(S.l). Some extensions and applications of Eq.(s.L') in machernatics stnristics

rwre d s o considered by Bczksalary and Styan (1993). But in ttlis thesis me orily consirter t he app1ication

of Eq.(S.l) to the reverse order law Bt-4t E { ( d B ) - ) . In addition. the results in Tlieorem S. 1 can also

I>r: extended to a product of n matrices. The corresponding results were presented L>y the üuttior i r i [94].

Theorem 8.2. Let -4 E Cm'"" and B E C"'P be giuen. Then

( a ) r [ ( - 4 ~ ) ( - 4 ~ ) t - (.4B)(Birlt)j = r [ B, -4*=IB] - r ( B ) = r(-4 * -4B - B B * . ~ ' = ~ B ). r 7

(b ) r[(.iB)i(-4B) - ( B ~ . ~ ~ ) ( A B ) I = r - r(-4) = 7-( ,4BB* - -4BB',-lt,-4 ).

(ci) (=LB)t(.lB) = (Bt.-Lt)(.4~) (=. ABB* = . - ~ B B * , L ~ . ~ R(BB8-4') C R(,-La) e Bt.4' c 1 (-4~)(12..1) ) -

(ct) The follozuing f o w statements are equiuulent:

( 2 ) ( - 4 ~ ) ( - 4 B ) t = ( - - l B ) ( B t ~ t ) und (-4B)t (-AB) = ( ~ t - 4 t ) ( - 4 ~ ) .

( 4 ) R(.-Ir.4B) C R(B) and R(BBœ-4') C R(-4').

Proof- Let LV = ,AB. Then by Eqs.(2.1), (2.7) and (1.8). it folIows that

4 ' B'BB' O 1

4 B'B

B S 4 * N'N

B'B Bœ.4".4B 1

i f i recl~lired for the first equality in Part (a). Applying Eq.(l.'Lj to it the block niatrk in it yiclcls tlic

scconcl equality in Part (a). Similady. we c m establish Part (b). The results in Parts ( c ) anci ( c l ) arc

direct consequences of Parts (a) <anci (b). The result in Part (e) follows directlu frorri Parts (c) ;trici ( c l ) .

The result in Tlieorem 8.2(e) is well known, see. e.g., Arghiriade (1967). Rao and Nitra (1971). Ben-

Israel nrid Greville (19S0), Campbell and Meyer (1991). Xow it can be regczrcletl is direct coriseqiicrices

of soine rank cqualities related to Moore-Penrose inverses of products of two matrices. n'e nest prescnt

:iriother ~ T O U P tank cqualities relateci to Moore-Penrose inverses of products of two niatrices.

Theorem 8.3. Let -4 E Cm"" and B f Cr'*" be given. Then

( a ) r f . 4 ~ ~ ~ - (AB)(rlB)t-4] = T [ B. .4*.AB] - r(B) .

( I I ) r-[=lt.4B - B(=IB) ' ( -~B) ] = r [ -aB* ] - ri-4)-

( c ) ~ [ ; L * = L B B ~ - B B ~ ; L * . ~ ] = &[B. .-Lm.4B ] - 2r(B).

(cl) r[.4+-4BB* - ~ ~ ' . 4 t . - l j = 2r [ ] - w - 4 ) -

( e ) (. .LB)(;LB)~A = .-lBBt o .4'.4BBt = BBtrl'.-l o R(.4*.-IB) C R ( B ) BT-4' C

( f ) -4 t .4~ = B(AB)~(..IB) ..L~.LBB' = BB'.-It.l t, R(BBœ.4') C R(.4*) o B'A' c { ( = ~ ~ ) ( l . - . ' l ) ) -

(c) The following stntements are equivalent:

Proof. We onfy show Part (b). Note that rlAt and (-4B)t(AB) are idempotent. W e have by Eq.(3-1)

that

The result in Theorem S.3(e) w~as established by Greville (1966). We nest consider ranks of rna-

t r k e-upressions involving Moore-Penrose inverses of products of three matrices. and then present their

c:onsequences relatecl to reverse order laws.

Theorem 8.4. Let -4 E CnLXnl B E C n x P and C E P X ' 1 be given. and let i\.f = .-LBC. Then

Proof. Applying Eqs.(2.Q) and the rank cancellation law (1.8) to Af - JI (BC)TB(,4B)t i \ f . \ u t obt,airi

(BC)*B(,-iB)* (BC)*(BC)(BC)' 0

r (AB)* (AB) (AB)*

= [ O (-4B)'Ji - i.(-4B) - r.(Bc)

O hf (BC)' - 111

(BC)*B(AB)* (BC) ' (BC) O

( - 4 4 (AB) ' O d.I

1 1 - r(-AB) - r( BC)

O ;CI - AI

Thus n-e have Eqs.(S.2) and (8.3). 0

Theorern 8.5. Let -4 E CnLXn, B E CnX" and C E CPXQ be given, and let 114 = .ABC. Then

(a) The rank of Mt - ( B C ) t ~ ( ~ - L ~ ) t is

(b) The following three s ta te~nents are equivalent:

(3) ,-LB B* BC = -4B(BCMt-AB) * BC.

(;LBc)' = (BC) 'B ( ;LB)~ and (.-LBc)~ = (B' BC)+ B ' ( ; ~ B B ~ ) ~ . iS.5)

Proof. Applying Eq.(2.12) to Mt - (BC)t B(-4B)t, we obtain

ici! * M hl * - W ( B ) ( B ) O

- B ) ( B ) M U (BC)' B(-AB)" O - r ( M ) - r(.-iB) - r(BC)

O O 0 (AB)=

O O (BC)' O

= ,-[ (BCJUB(-dB)* (BC)'(BC)M * 1 - r (M) A l ' ( .AB) (-AB 1 * !\f'A.lM'

c2$ reqiiired for Eq.(S.4). Then the equivalence of Statements ( 1 ) and (2) in Part (b) follows immetlintely

frorn Eq.(8.4), ancl the equivalence of Statements (2) and (3) in Part (b) foltows frorn Lenima 1.2(f). If

I - ( - ~ B C ) = r(B)? then

On the other hand.

- r (.AB) (AB)' Ad -ABB*BC

Thus we have r 1

Thus according to Statements (1) and (2) in Part (b) , we knoix

truc. The second equality in Eq.(8.5) follows from writing -4BC =

cquality to it. 17

that the first equdity in Eq.(S.S) is

.-lBBtBC and tlieri app1ying clle tirst

Theorem 8.6. Let -4 E CmXn. B f CnXP and C E C P X Q Ue given. und k t L'Li = .-IBC- Then

Proof. Applying Eq.(2.11) to B+ - ( A B ) ~ L v ~ ( B c ) ~ , we obtain

as requirecl for Eq. (8 5). 0

Theorem 8.7. Let -4 E C n t X n , B € Cnxp and C E CPx' I I , git~en. Then

( a ) ABC)^ = ( . ~ ~ - ~ B C ) ~ B ( = L B C C ~ ) ~ .

(ù) (ABC)* = [(;~B)+..IBc]~B~[-.IBc(Bc)~]~.

( c ) ( C h e , 1964) ( . 4 ~ ) t = ( -4 tAB) t ( -4B~t ) t .

(ci) If R(C) E R[(.4B)'] und R(Aœ) C R(BC), then (-4BC)t = Ct~t , - l t .

Proof. Write ,ABC as -4BC = - ~ ( - ~ ~ A B C C + ) C . Then it is evident that

r(-4t--i~cC't) = +WC), lZ[(-4l3CCt) '1 C R(C). and R[((;L' .-IBc)~ )'] C R(;L').

Tlius Ly Eq.(8.5), we find that

as reqiiirecl for Part (a). On the other hand, we can write -4BC as -ABC = ( . ~ B ) B ~ ( B C ) . Applying thc

tquality in Part(a) to it yields

as recpired for Part (b). Let B be identity matriv and replace C by B in the result in Part (a). Tlien u-e

have the result in Part (c). The two conditions in Part (d) are equivalent to

111 ttiat case. the resuit in Pa r t (b ) reduces to the result in Part (ci). 0

Tri the remaincler of this chapter we consider the relationship of (-ABC)t and the reverse orcier procliic:r

C i ~ t . 4 t , and present necessary and sufficient conditions for (-ABC)+ = CtBt.4' to Iiold. Some of rhr

rcstilts were presented by the author in [SI] and [93].

Lerrima 8.8[93]. Suppose that --Il, .A3, -A3, Bi und B2 satisfij the the following range inclusions

Proof. The range iriclusions in Eq.(S.7) are equivdent to

Ir1 t h t c~îse, it is easy to verify tha t the block mat& in the rigtit-hand side of (8.8) and the giwn tdock

riiatris in tlie left-hand side of Eq.(S.8) satisfy the four Penrose equatioris. Thus Ecl.(S.S) holds. G

Lemma 8.9. Let -4 E C m X " . B E C n x P and C E Cl'"" Oe given. Then the p ~ û d ~ ~ c t C+ ~ t . 4 ~ crm be

where the bluck matrices P, J and Q satisfy

r ) = ( A ) + ( B ) + ( C ) R(QA) C R(J), and R[(-4P)'J R(.J'). ( S . IO)

Proof. Observe that

R(B'-A*) R(B'BB'), R(-4B) Ç R(,4,4*), R(C*B8j C R(C*C), R ( B C ) C R(BB* B ) .

as well as the three basic equalities on the Moore-Penrose inverse of a nirrtrix

lv; = ~ V * ( N * ! V N * ) ~ N * . LV~ = ( N - N ) ~ N = . lvt = L~*(L\rLv=)t.

Thus ive find b - Eq.(S.S) that

I 0 O -4-4 * ( C C ) + C * B'(BB' B)tB'.-l'(,4,4')t

II BmYi3* D - l ] = [ * C'C C'B' O * O O

Herice we have Ec[.(S.9). The properties in Eq.(S.10) are obvious. 0

T h e o r e m 8.10. Let -4 E C'""", B E C n X P and C E CpX'1 be given and let J I = .ABC. Then

1 O BB'B BC 1 = r ( d ) + r ( B ) + r(C) - r ( M ) . (S. 12)

Proof. It follows from Eqs(l.7) and (5.0) that

as recluired for Eq.(S. 11) - O

Theorem 8.11. Let -4 € CrnX ". B E C n X p and C E C"'1 be giuen and let :'CI = -4BC. Then

Proof. Notice that

IlTc. iïrst get the following

Observe from Eq.(S.IO) that

Thris nTe find by Eq.(1.7) tha t .

( [ - " * ~ ~ - ; ] ~ [ ~ ; : j ) 1- [CLW' C P ]

The resuIts in Eqs.(S.I3) and (8.14) folow frorn it. 0

C'C C'B* O

CAP' [ O C O O O O l

Corollary 8.12. Let -4 E C m x n , B E C n x p and C E C P X Q be giuen, rrnd suppose thrrt

- - r

R(B) c R(-4') and R(B*) C R(C).

( S . 16)

- f f * i O O O - O

O O O O -4

O O B'BB* B* -4" 0

-C*CM' O C*B* O O

O C O O il - d

( ; ~ B c ) ~ = ct~+--Lt o R(.-I*-dB) C R(B) and R[(BCC*)*] C R(B*) . (S. 17)

- r ( M ) - - r ( . J )

Proof. Eq.(S.lS) is equivalent to .4t.4~ = B and BCCt = B. Thus rve c m recluce Eq.(8.13) by I h c k

elerneritary operations to

-BCL\I*,~B 1 BCC'

O BB'B B 1 - 21-( 5 )

-4' -4B B O

O BCC'B* BCC'

O BB'B - 2r(B)

-4*-4B B O

B

= 1 BCC* 1 +r[B. - - IR-AB]-2r (B) .

Corollary 8.13. Let B E Cm"" be giuen, -4 E CnLXm and C E C n X n he two inuertible nrutnccs. Let

Proof. Follows immediately from Corollaq 8.12.

Theorem 8.14. Let B f Cmxn be giuen, A E C7'1Xn' and C E C n x n 6e two irtvcrtible ~mcrtrices. Let

:LI = --WC. Then

(a) r[MfLlt - - - i ~ B ~ - - l - ' ] = r [ B , ..l'AB] - r ( B ) . - - ( 1 1 ) r [ Mt Al - C L B t B C ] = r

B 1 BCC. 1 - r ( B ) .

Applying Eq.(-Ll) to both of them yields the results in the theorcm. Q

Chapter 9

Moore-Penrose inverses of block matrices

1x1 this chapter we wish to establisfi some rank equalities related CO the factorizations of 2 x 2 block matrices

and ttien deduce from them various expressions of Moore-Penrose inverses for 2 x 2 block matrices. as

well as m x TL bImk matrices. Most of the resdts in this chapter appears in the author recent paper [95]. r 1

-4 B Theorem 9-1. Let -4 E C m X n . B E Cm"". and C f C L X " be giuerr. a n d factor il1 =

I I I ~ L W ~ : Eo = Ini - BBt and Fc = in - Cf C. T f t e n

(a) T h e runk of il/[' - Q - ~ L V ~ P - ~ satisjies t he e p a l i t y

r 1

(b) Tf l e Moore-Penrose inverse of hi cnn be expressed us Mi = C ) - L i V t ~ - L . tlrrrt is.

fiulds if and on ly if -4. B and C satisfy the rank additivity condit ion

Proof. It follows first by Eqs.(9.1) and (8.19) that

Tlic ranks of the two block matrices in (9.6) can simplify to

Piitting both of them in Eq-(9.6) yields Eq.(9.2). Notice that ( ~ ~ - 4 F ~ ) t B t = O arid ~ ~ ( E u - - l F c ) ~ = 0

aitvays hold. Then it is easy to verify that

Putting it in Eq.(9-2) and letting the right-hand of Eq.(9.2) be zero. ive get Ecp(9.3)-(9.5). @

Corollary 9.2. Let -4 E C n L X n , B E cmX%nd C E C l x n I,e giuen. lf

R(-4) n R(B) = ( O ) and R(A*) f~ R(C') = { O ) ,

Proof. Under Ec1.(9.7), the rank equality (9.4) naturally holds. In t h a t case. ive know by Tlicorern 7.8

that - - I ( E ~ = L F ~ ) ~ - ~ = -4. Thus Eq49.3) reduces to Eq.(9.S).

Theorem 9.3. Let -4 f C"'Xn: B Cm' "" and C E CL r1 be giuen. Tften

Proof. It is eüsy to verify that

trllcrc P nrld (2 arc rionsingular. Thcn Ive have by Eq.(S.5) tliat

\kitteri in an esplicit forrn, it is Eq.(9.9). Now let ,Y = Bt-4 and k' = .4Ct in Eq.(9.9). Thcri Eq.('3.9)

Sote tiiat

Then it follows bv Theorem 7.9(e) and (f) that

and

Thus w-e have Eq.(9.10).

Theorem 9.4. Let -4 E CmX ", B E CmX" C C Clxn and D E Cf X n be giuen, and factor M =

( a ) The rank of - &-' i ~ f P-' satzsfy the equality

(a) The Moore-Penrose inverse of LU in Eq.(9.11) can De expressed as h1t = Q-':VTP-' . t f ~ t as.

holds if und only if -4, B, C and D satàsfg

Proof. IL follows by Eqs.(S.ll) and (8.19) that

The ranks of the two block matrices in it can reduce to

Sirnilady we can get

Thus we have Eq-(g.E), Eqs.(9.13)-(9.15) are ( iirect consequences of Eq. (O. 12

CoroUaxy 9.5. Let -4 E C m X n , B E cmXk und C E C l x n be gauen. and factor 31 = 1 c : 1 ( ~ - 5

pr~rticular. the Moore-Penrose huerse of Ad in (9.16) can be eqressed us

i f m d onlg i f -4, B und C sutisfy the following runk udditivity condition

Clearly iV in Eq.(9.11) can be written as

Thcn it is easy to veri@ that

Thus if we can find i\i!, then we can give the expression of N t in Eq.(0.21). This consideration motivitcs

us to find the following set of results on Moore-Penrose inverses of block matrices.

Lemma 9.6. Let A E Cm"", B E cmxk, C E clXn and D E CL"" be giuen. Then the r m k mlditivity

is e p l v a l c n t t o the two range inclusions

(md the rcrnk additiuity condition

Proof. Let

R(Vl > n R(VJ = { O ) and

Iri that case, we easily find

hoth of wliich are equivalent to the two inclusions in Eq.(9.23). On the otl-icr hand. observe that

Thiis according eo Eq.(9.26) and Lemrna 1.4(b) and (c) , we find that

~Frorn both of them and the rank formulas in E q ~ ~ ( l . 2 ) . (1.3). (1.5): (9.23): (9.27) and (9.28). we derive

the following two equalities

Both of them are esactly the rank additivity condition Eq.(9.24). Conversely. adcling r(-4) to the three

sitfes of Eq.(0.24) and tlien applying E q ~ . ( l . 2 ) ~ (1.3) and (1.5) to the corresponding resrdt n-e first obtain

E.4 B = r [ A . E.4B]+r[CF--l . S..,]. (9.29)

On the ottier Iiand, the two inclusions in Eq.(9.23) are also equivalent to

-4 B -4 B -4 B -4 -4 B O

C D 0 C D C

-1pplyiiig Ecl.(l.5) to the right-hand sides of the above two eqiialities and thcn corribining theni n-itti

Eq. (9.29), n-e fincl

-4 Ed4B O I-(nr) = T = ~ [ . 4 , E A B ] +T-[CF..~. SJ. C ] = r-1-4. BI + r[C. DI. CFA s.4 c

Both of them are esactly Eq.(9.22).

Siniilarly we can establisli the folIowing.

Lemma 9.7. The rank additiuitg condition (9.22) is equivalent to the following f o u r conditions

Theorem 9.8. Suppose that the block matrix M in Eq.(9.11) satisfies the rank additivity condition

(!3.22), then the Moore-Penrose inverse of M can be expressed à71 the two f o m s

( 4 ) .it (C)

Proof. Lemma 9.6 shows that the rank additivity condition in Eq.(9.22) is eq i i iden t to Eqs.(9.23)

ancl (9.24). It follows from Theorern 9.4 that under Eq.(9.23). the Moore-Penrose inverse of 31 cari

I E expressed as Eq.(9.13). On the other hand. Et follom-s from Theorcm 9.1 that iinder Eq.(9.24) tlic

1loortiPenrose inverse of 1V3 in Eq.(9.20) can be written as

where BI = E,\B. CL = CF4 and J ( D ) = Ec,S,tFs,. Sow siibstitiiting Eq.(9.34) into Eq.(t).Zl) arid

tiicri Eq.(9.21) into Eq.(9.13). we get

Writtcn in a 2 x 2 block matriu. Eq.(9.35) is Ec1.(9.32). In the sarne tvay. ive can also riecornpose JI i11

Eq.(9.11) into the other three forms.

Bcased on the above decornpositions of Ad we can d s o find chat under Eq.(9.22) the Moore-Penrose inverse

of h l c m also be expressed as

Finally from the uniqueness of the Moore-Penrose inverse of a matris and the expressions in Ecls.(9.32)

:mcl (9.36), we obtain Eq.(9.33). El

Souie fundamentai properties on the Moore-Penrose inverse of ILI in Eq.(9.11) c'an be derive frorrl

Eqs.(9.32) and (9.33).

CoroIlary 9.9. Denote the Moore-Penrose inverse of A l in Eq.(9.11) 6y

L A

~ U ~ L C T C CI. G2! G:$ und G.t m e n x 7n. n x i, k x 7n and k x 1 nratrices. respcctively. If J I in Eq.(9.11)

satisfies the runk additiuity condition (9.2'2): tlten the subrnatrices in h l a n d :Ut sutisfg tltc rank ct~ua1itie.s

w f ~ e w b; , I.>. und are defined i n ( 9 .Xi).

Proof. The four r m k cqudities in Eqs.(9.38)-(9.42) c m directly be deriveci frorn the cxprcssiori iri

Eq.(9.33) for hf arici the rank formula (1.6). The two equalities in Eq.(9.42) coirie from the siirns of

Eqs.(9.38) and (9.41). Ecls.(9.39) and (9.40), respectively. The two results iri Eq.(9.43) arc derivccl frorn

Eq.69.22) and Theorem I.lG(c) and (cl). 0

Tlic r a i k additivity condition Eq.(9.22) is a quite weak restriction to a 2 x 2 block matrix. .As rriatter

of fact, ruiy matrix satisfies a rank additivity condition as in Eq.(9.22) when its ron-s and coluriins are

properly perrnuted. We next present a group of comequences of Theorem 9.8.

Corollary 9.10. If the block riaut* Ad in Eq.(9.11) satisfies Eq.(9-23) und the following two corrditions

R(CL) R(&) = { O ) and R(B;) n R(S;\) = { O ) , ( 9 .44

then the Moore-Penrose inverse of Ad can be expressed us

~ l ; h e ~ e Cl, BI . H z , H3 und J(D) are as in (9.32)? P und Q are us in (9.13).

Proof. The conciitions in Eq.(9.44) irnpIy that the bIock mat rk 1V2 in Eq.(9.20) satisfies the follon-ing

rartk atlditivity condition

r (NZ) = r(E.4 B) + r ( C R i ) + r(S,i),

ivhich is a specid case of Ecl.(0.24). On the other hand. under Eq.(9.44) if follow by Thcoreni 7.7 that

S,i.Jt(D)S,i = Sri. Tlius Eq.(9.35) reduces to the desired result in the corollary. O

CorolIary 9.11. If the block matr iz M i n Eq.(9.11) satisfies Eq.(9.23) and thc two contlitio~rs

R(BS,) c R(.-l) und R ( C S J ) R(--Lw). (9.45)

tlren the Moore-Penrose inverse of IV cun be expressed as

Proof. Clearly Eq.(9.45) are e q u i d e n t to (EclB)S.i = O and S:,(CF..t) = O. In t h t case, Ec1.(9.24) is r 1

satisfiecl, anci Nt = 1 ( E 2 ) t (Cz)t 1 iiiEq.(9.35). [7

L -1

CorolIary 9.12 (Chen and Zhou. 1991). If the block 7nutriz in (9.11) satisfies the fo1fozu.rri.q f u ~ r r

tiien the Moore-Penrose inverse of hl can be expressed as

Proof. It is easy to verify that under the conditions in Eq.(9.46). the rmk of AI satisfies the rank r 1

additivity condition (9.22). In that case, N t = 1 " Ot 1 in Eq.(9-35). 0

0 S., L -1

Corollary 9.13. If the block matrix M in (9.11) satisfies the four conditions

R(.4) n R(B) = {O), R(.4*) n R(C*) = (O). R ( D ) R(C) . R ( D ' ) c R ( B œ ) . (9.47)

then the Moore-Penrose inverse of iV can be expressed as

wherr S.-, = D - C-it B. BI = Ee4B and Cl = CFzt.

Proof. It is riot tlifficult to verify by Eq.(l.5) that under Eq-(9.47) the rank of :ll s~tisfies Ec1.( l ) .2) . In

t h t c~lse. J ( D ) = O =

Corobry 9.14. If t f ~ e block matriz fkf in (9.11) satisfies the four condztzo~ts

( 4 ) R B ) = O } R(--iœ) n R ( C ) = {O) ,

then the Moore-Penrose inverse of kf can be expressed as

Proof. Clearly Eq.(9.49) is etquivalent to Ctsa = O and s . ~ B ~ = O , as well as S.:~C = O and BS: = 0.

Frorn tliern and (9.48). we also find

Combining Eqs.(9.48) and (9.50) shows that Ad satisfies Eqs(9.23) and (9.24). 111 tliat case.

in Ecl-(9.35). 13

Corollary 9.15. If the block matrix k1 in Eq.(9.11) satisjies the rank additiuity co~rdition

r ( M L ) = r ( A ) + r ( B ) + r ( C ) + r ( D ) , (9.51)

then the Moore-Penrose inverse of ibf can be ezpressed as

Proof. Obviously Ec1.(9.51) is a special case of Eq.(g.Z) . On the other Iiand. Eq.(9.3L) is aIso equivi~lent

t o the follonring four conditions

R(.-i) f7 R(B) = {O), R ( C ) n R ( D ) = {O): R(_4') n R ( C * ) = { O ) . R ( B * ) n R ( D ) = ( O ) .

Iri that case.

( 4 ) = R ( 4 ) ( ) = ( 4 ) . ( 4 ) = 4 ) R(&) = R(B) .

R(C,) = R(.4), R ( G ) = R(Ca) , R ( D ; ) = R(Dm), R(.4?) = R ( D ) ?

by Lcmma 1.2(a) and (b). Tlien it turns out by Theorem 1.2(c) arid (ci) tha t

4 4 = 4 . 4 4 4 = 4 . B: B~ = B~ B. LI2 B = B B'.

c~c:=cc~. c.@=c~c. D ! D ~ = D ~ D . D , D : = D D ~ .

Tbus Eq.(9.33) reduces to Eq.(9,52) 0

Corollary 9.16. If the Dlock matrïx hf in (9.11) satisfies r ( M ) = r(.-L) + r ( D ) u n d

then the Moore-Penrose inverse of Ad c m be ezpressed as

Corollary 9.17. If the lrlock rnatrix Ar[ in Eq-(9-11) satisfies r ( M ) = r(,4) i- r ( D ) and the jollowing

four CO 72 ditions

then the Moore-Perwose inverse of 114 can Oe expressed as

The above two corollaries c m directly be derived from Eq~~(9.32) and (9.33). the proofs are omitted

Liere.

\lrithout mucli effort, we can e-xtend the results in Theorem 9.8 to m x n block matrices when they

satisfy rank additivity conditions.

Let

t)e ari 7 r ~ x n block rnatrix , mhere -4, is an si x t j matri.. (1 5 i 5 m, 1 5 j 5 r ~ ) . and suppose that 31

satisfies the following rank additivity condition

For convenience of representation, we aciopt the following notation. Let = (-4;;) be given in Eq.(9.53).

wtiere .qij E C s g X t ~ 1 5 i 5 rn, 1 5 j < n, and Cr.., si = s, x:=, ti = t . For eacli .dij in 31 ive associate

ttirce block niatrices as follows

The syrnbol J(-4;j) stands for

ivhere ai; = B i j F D i j , fii; = EoijCij and Soi, = -Aij - is the Schiir cornplement of Dij iii JI.

We cal1 the matris J(Ai,) the runk complement of -4, in hl. Besides me partition the Moore-Penrose

inverse of ibf in Eq.(9.53) into the form

ilfi =

wherc Gi j is a ti x sj rnatrk. I 5 i < n, 1 5 j 5 m-

Nest we build two groiips of block permutation matrices as follows

wherc 2 5 i 5 in. 2 5 j 5 n. Appiying Eqs.(9.61) and (9.62) to LU in Eq.(9.53) and Ut in Eq.(g.GO) ive

have the following tmo groups of expressions

TIwe two cclualities show that ive can use two block permutation matrices to permute .4,, in -11 arid

the corresponding block G,; in M i to the upper left corners of .bI and Mt, respectively. Observe tiiat

Pt and Q j in Ecp(9-61) and (9.62) are al1 orthogonal matrices. The Moore-Penrose inverse of P,MQ, in

Eq.(9.63) can be expressed as ( P ~ L L I Q ~ ) ' = Q T M + P ~ . Combining Eq.(9.63) with Eci.(D.64). ive Ilaie the

following simple result

If the block matrix Ad in Eq.(9.53) satisfies the rank additivity condition (9.54). then the 2 x 2 block

rriatris on the right-hand side of Eq.(9.63) naturally satisfies the foliowing rank additivity coridition

h e r e 1 5 i 5 r n , 1 < j 5 n. Hence combining Eq~~(9.65) and (9.66) with Thcorenis 0.8 and 9.9. 1vc

obtain the following general result .

Theorem 9.18. S7~ppose that the m x 71 lrlock matriz h1 in Eq.(9.53) satisjies tfre rrrnk ntiditit.itg

condition (9.54). Then

(a) The Moore-Penrose inverse of !\I can be ezpressed as

w f ~ e r e .J(.-lij) is rfefineù in Eq.(9.59).

(1)) The mnk of the block e n t q G j i = J t (=lv) in M t is

al+ M = diag( bit vL. . - - - . 1;,! 1.; j.

mi t t en in e q l i c i t jorms: Eqs.(9.69) and (9.70) are equivulent to

Ir1 addition to the expression given in Eq.(9.67) for M t . wc can also cierive some otlier espressioris for

Gij in M t from Eq.(9.32). But they are quite complicated in form. so Ive omit tticra Iiere.

Various consequences c m be derived from Eq-(9.67) iVhen the siibrnatrices in AI satisfies sonie aci-

clitiorial conditions. or 111 lias some particular patterns, such as trianguiar forms. circulant forms and

txiclirigonal forms. Here we only give one consequences.

Corollary 9.19. If the block matriz i\f in Eq.(9.53) safisfies the followiny rank uddztiuity co~rdition

then the Moore-Penrose inverse of h.1 can be expressed as

rrrherc Bi j rrnd Ci j are defined in Eqs.(9.56) and (9.57).

R(Ci,) n hl(Dij ) = {O}; R(Bz;) n R(D,) = {O} : 1 5 i 5 in. 1 5 j 5 n.

We can from them . J ( = l i , ) = EBi, Ai, Fctl - Putting them in Eq.(S.GI) yields Eq.(9.'7'L). 13

Chapter 10

Rank equalities related to Moore-Penrose inverses of sums of matrices

In this chapter. LW establish rank equalities related to Moore-Penrose inverses of sums of matrices and

corisitier their various consequences.

Theorem 10.1. Let -4, B E Cr""" be given a n d let 1V = -4 +- B. Then

.dt + of E {(A t 8) - } c=, r ( A + B ) = r (A) + r (B) - r(.4BS.-L) - riB.4'B). ( 10.2)

Proof. It follon-s by Eq(2 .2) and bIock elementary operations that

-4 - -4*4 - O -4 = N

r[iV - N( --If + B T )IV] = O B'BB* B*N - r(-4) - r ( B )

1v *4* h-Bo 1 Thiis we have Eqs.(lO.l) and (10.2). 0

A generd result given below.

Theorem 10.2. Let -Ar, .4.r: - - - , E C m x n be giuen and let -4 = .AL + + - - - + -Ak. -y =

.-If + -4: + - - - + -4;. Tlren

Proof. Let Pl = [ I n , - - - , I n ] andQ1 = [ L m , - - - , I n ' ] - . T h e n S = PIDtQ, . In that case. it follou-s

by Eq.(2.1) that

= I'( DD* D - DP;.-I'Q;D ) + 4.4) - r ( D )

as required for Eq.( 10.3). 0

Theorem 10.3. Let -4, B E C m X n bc given und let N = -4 + B. Then

Proof. Follows irnmediately From Eq. (2.7). O

It is well known that for any two nonsinguiar matrices -4 and B. there aiways is .4( -4-' + 5-1 ) B =

-4 + B. Yom for Moore-Penrose inverses of matrices we have the FolIowing.

Theorem 10.4. Let -4. B E C m X n be given. Then

.4( -4' + B~ )B = -4 + B o A+(-+I+ B)B+ = + B~ o R(.L) = R(B} and R(As) = R(B ' ) . (10.9)

-4 + B - 4 - 4 ' + B + ) B = -4 + B - [ A . -41

aricl theri applying Eq.(3.1) to it produce Eq.(lO.ï). Replacing -4 and B in Eq.(10.7) respectively by -AT

and Bt leac-ls to Ecl.(lO.S). The equivaiences in Eq.(10.9) follotv from Eqs.(lO.ï) ancl (10.8). 17

Theorem 10.5. Let -4, B E C n L X be giuen and let 1V = -4 + B. Then

Proof. Let P = Es,4FB and Q = EriBFri. Then it is easy to verify that

P'B = O. BP' = 0, Q'.4 = O. .4Q' = O

and

P* PP* = P*--IP*, Q'QQ' = Q'BQ'.

T h s w e find by Eq.(2.2) that

P'*-lP* O

= .[ O Q*BQm

4 BQ' A i - B

where

by Eq.(1.4). TIius we have Eqs.(lO.lO) and (10.11). 0

Theorem 10.6. Let -4, B f Cm"" be giuen. Then

( a ) ~ ( ( . ~ + B ) ( ; L + B ) ~ - I d , B ] [ A l B I i ) = rirl. B ] - 1 - ( - 4 i B ) .

( c ) ( - 4 + B ) ( - 4 + ~ ) + = [ A , B][--l' B ] t * r[-4, BI = r ( - 4 + B ) b R(-4) C R ( - A + B ) (mi

R ( B ) c R ( A + B).

Proof- Let LV = -4 + B and 111 = [.A. B 1. Then it follows from Theorem 7.2(a) ttiat

as recpired for Part (a). Siniiiarly we have Part (b).

In gcneral we have the following.

Theorem 10.8. Let -4. B E C l n X n be giuen and let !V = -4 + B. Then

(a) r( -4AÏtB) = r ( A r - A * ) + r (B*N) - r ( N ) .

( ) ( - 1 ) = ( 4 ) -i- ( B ) - ( N ) zf R(-4') c R(i\rg) r ~ r ~ d R ( B ) R(Z.) .

( c ) r( ..LIV~ B - BN+-4 ) = 7- [ lc- ] +r[!V, -4.- ] - 2 r ( f i ) .

(TL particuLar~.

(c l ) .A-W~B = O r ( N - 4 * ) + T ( B * N ) = r (N) .

( e ) , L N ~ B = ~ N t . 4 (=, R(A1V') C R ( N ) and R(-4'N) C R ( N * ) .

( f ) - 4 N t ~ = BIV~-J . if R(A) R ( N ) und R(il') C R(N*) .

Proof. It follows by Eq.(2.1) that

N*.W* + N œ B N ' W B - r ( N )

AN* O

1V' B

= r [ ] - r ( N ) = r-(Nd-lw + r ( * * ~ v ) - r(.N).

as reqiiirecl for Part (a)- Lnder R(.-l') C R ( N 9 ) and R ( B ) C R(:Y). wr linon- that r(iV.4') = r(-4) :wci

r.(B9!\-) = r ( B ) . Thus WC have Part (b) . SimiIarIy it follows by Ec1.(2.1) that

1 ;INa BN* O 1

-4s rccliiired for Part ( c ) . D

It is well known that if R(A8) Ç R(1V8) and R(B) C R ( N ) , the product -4( -4 + B )t B is calletl the

parallel sum of -4 and B and clenoted by P(A, B). The results in Tlieorem 10-S(b) ancl ( f ) sliow tliat if

-4 arlcl B are parallel summable. then

r - [ P(,4. B ) ] = r(.4) + r ( B ) - r( -4 + B ) and P(.-l. B ) = P ( B . -4).

These two properties were obtained by Rao and Mitra (1972) with a different methocl.

Tlie following three theorems are derived directly from Eq.(2.1). Their proofs are oniittccl liere.

Theorern 10.9. Let A, B € Crnxn be yiuen. Then

Tlic equivalence in Eq.(10.13) was established by Marsaglia and Styan(lSï-4).

Theorem 10.10. Let --Il, -A2, - - - -Ak E Cmx be giverr und denote

The eqiiivalence in Eci.(lO. 14) \vas established bÿ 'vlarsaglia and Stpn(l97.1)-

Theorem 10.11. Let -4, B E C n L X n be yiven and let IV = -4 + B. Then

(a) r( -4 - --LVt.4) = r(NB*iV) + r(-4) - r ( N ) .

(b ) r( -4 - -4iVt.4 ) = r(.-L) + r(B) - r (N) , if R(=l) C R(iV) mtd R(=ls) C R(iV-}

(c) Nt E {-A-) u r! iVB*N) = r ( N ) - r(-4).

( c l ) Nt E { -A- ) if r ( N ) = r(..l) + r(B).

In the rernainder of ttiis chapter, we present a set of resrrIts relatecl to cspressioris of llr)oi-c~-Pc~~ir.osi~

iiiverses of Schur complements. These results have appeared in che autlior's rescerit papcr[%].

Theorem 10.12. Let .4 E B E C m X k , C E C ' X f l and D E CL" " be yivcm. Thcrr the rmrk ndrlitiuit!/

therr the following inversion formula holds

where

S A = D - C A ~ B . S D = - 4 - B D ' C . J ( D ) = E c , S n F e , .

Proof. Follows irnmediately from the two expressions of M t in Theorern 9.5. 13

The results given below are al1 the special cases of the general formula Eq-( 10.17).

Corollary 10.13. If -4. B. C and D sutisfy

und the foliowing two conditions

or more specific~~lt~/ sdisf?y the four conditions

tlren the Moore-Penrose inverse of the Schur complement SD = -4 - BDtC sotisfies the inversion fot - rnd~

iuhere SA, B1 , CI and J ( D ) are rlefined in (10.17).

Proof. It is obïious ~ h a t Ecl.(10.19) is equivalent to (EDC)S;I = O and S;(BFD) = O. or ec~riivalcric1~-

These two equalities cleady imply that SD, EoC and BFD satisfy Eq.(9.31). Hcnce by Lemnia 9.7. WP

krlow tliat under Eqs.(lO.lS) and (10.19). -4. B, C and D naturally satisfy Eq.(lO.lG). Son. siibstituting

Ecl.(lO.22) into the left-hand side of Eq.(10.17) yields . ~ t ( - 4 ) = (-4 - B D ~ C ) ~ . Hence Ec1.(10.17) becornes

Eq . ( l02 l ) . Observe that Eq.(10.20) is a special case of Eq.(10.19). herice Eq.(lO.Zl) is also truc iiridcr

(10.20). O

Corollary 10.14. If -4. B, C urtd D satisfy Eqs.(10.18). (10.19) ( r d the foflowinq two conditior~s

tuhere SA, B I , CL and J ( D ) are defined in Eq.(10.17).

Proof. =lccording to Theorem 7.7, the two conditions in Eq.(lO.'Zl) irriplies t h Stl J t (D)S,q = S..\.

H c r m Eq.(IO.21) is simplified Eq-to (10.24). 0

Corollary 10.15. If -4, B. C und D sutzsfp Eqs.(lO.l8). (10.19) und the following two conJitio7r.s

R(BS,) c R(.-l) und R(C'Sli) C R(.4')? ( 10.25)

where S..\ = D - C-4' B.

Proof. Clearly. Eq.(10.25) iç equivalent to ( E L B)S:, = O zinc1 S*,(CF..\) = O. ti-hich c m also i - ~ c ~ i i i ~ . ; ~ l ( ~ r l t l \ -

tw espressed as S,i(E-4B)t = O and (CF.-l)tS.4 = O. In that case. . f ( D ) = Ec., S..\FB, = I3trric:c

Eq.(10.21) is simpiified to Eq.(10.26). 0

Corollary 10.16. If -4. B , C nnd D satisfy Eqs.(lO.lS). (10.19) and the foLlowi7r!l tzuo contfitio.rr.s

R(B) Ç R(.4) und R(C') E R(.4*), ( iO.27)

Proof. The two iriclusions iri Eq.(10.2?) are equivalent to E,\B = O and CF_I = O. Substitiiting thcrri

into Eq.( 1021) yielrls Eq.(10.28). O

Corollary 10.17. If -4, B. C und D sutzsfy the following four conditions

Proof. Gnrler Ecp-(10.29) and (10.30), -4, B, C and D naturaliy satisfy the rank additivity c:oricIition

in Eq.(10.16). Besicles, from (10.29): (10.30) and Theorem 7.2 ( c ) and (d) we c;tn dcrive

Siit)stituting thern into Eq.(10.17) yields Eq.(10.31). [7

If D is invertible, or D = 1: or B = C = -D. then the inversion formula (10.17) cal1 retliicc t a sorne

other simpler forms. For simplicity. we do not list them here.

Chapter 11

Moore-Penrose inverses of block circulant matrices and sums of matrices

Let C bc a circiilant niatris over the coniplex number field C with the form

Tlieti it is a well known result (see, e.g.. Davis, 1979) tha t C satisfies the follon-ing faccorizaciori eqilnlity

Cr*CU = diag(X1, A-, - - - , Ar, ). (112)

wtierc b- is a1 unitary matrix of the form

Tt is eviclent that the entries in the first row and first coIurnn of U ;ire a

Otiservc that Ci in Ey.(ll.3) 11as no relation with ao-ak-~ iri Ecl.(ll.l). Thus Ecl.(l 1-2) cari directiy Iic

cstencled to block circulant matris as follows.

Lemma 11.1. Let

be a block circulant mat& over the complex nuntber field C , where .4[ E CnZX", t = 1.

.strtisfics the following factorization equality

G.4Un = diag( J I , J2, - - - . Jl; ),

whcre Cr, and Us ure two Iiloclc un i tap j matrices

Uni = ( ~ ~ ~ 1 n r ) k x k r L'.ri = ( ' l ~ p q ~ n ) k x k 7

100

Especicrllg, the block entries in the Jirst block rows and first block columns of O',, c~nd U,, are d l scrrlnr

pmclwts of 1/Jk with irientity matrices. and Ji is

Observe that JI in Eq.(11.7) is the sum of ,AI : -4?: - - - . -dk. Thus Eq.(ll.T) irnplies that the suni

= 4 is closely linked to its corresponding block circulant marris through a uriitary fücrorizarion

cquality- Recall a firndarnental fact in the theory of generdized inverses of matrices (see, e-g.. Rao and

lSlitra, 1971) that

( p - 4 ~ ) ~ = ~ ~ - 4 ~ P*' if P and (Z are unitary. (11.21)

Theri from Eq.(11.7) we can directly find the following.

Lemma 11.2. Let -4 be given in Eq.(l l.G), Ur and C', Oe given i 7 ~ Eq-(11.8) - Tlren the Moore-Pen7-ose

inuerse of -4 sutisfies

u;-4'Urn = =diag( J:. 4, - - - . .IL 1. (11.12)

Proof. Since U,,, and Un in Eq.(ll.T) are unitary, we find by Eq.( l l . l l ) that

(br;*4un)t = u;-4tun1*

On the other hand, it is easily seen that

[diag(Ji; Jj: - - - . . J k ) I t = diag(.l/. .I!. - - - . .JL ) .

Thus Eq.( l l . l2) follows. O

A main consequence of the above result is presented below.

Theorem 11.3(Tian, 1992b, l998a). Let ..Il, -4-, . .-Ir, E CnaX ". Then the Moore-Pe7aro.s~ ZTLZ~CT.SC of

t h i r sum satisfies

Jn par-ticular. if the block circulant m a t e in it is nonsingular. then

Proof. Pre-multiplying [In, 0, - - -. O] and post-rnultiplying [ I , , 0. - - - . U I T on the botli sides of

Eq. ( il. 12) inimediately yield Eq.(11.13) - [7

It is eâsily seen that combining Eq.(11.13) with the results in Chapter 9 may produce lots of formula

for the Moore-Penrose inverses of matrix sums. We start %<th the simplest case-chc Moorc-Penrose

inverse of sum of two matrices.

Let -4 and B be two rn x TL matrices. Shen according to Eq.(11.13) ive have

Ar; a special ccase of Eq.(ll .lS). if we replace -4 + B in Eq.(l l.15) by a co~nples rriatris -4 + iB. n-hcrc

t~otli -4 and B are reai matrices, then Eq.(11.15) becorries the ecluülity

Xow applying Theorems 9.8 and 9.9 to Eqs.(ll.lB) and (11.16) we find the t'ollowi~ig two resiilts.

Theorem 11.4 (Tian. 1998a). Let -4 and B Oe two m x n contplex matrices. and suppose tfrrrt Lht?lj

srrti.sf?/ the rnnk udditiuit~/ conditilin

o r r~lterr~utively

l?(+4) ç R(-4 & B) and R(.4*) C R(,4' k B') .

T f w n

(a) The Moore-Penrose inverse of -4 + B can be q r e s s e d as

-4 B r l~ f~ere .J(.4) und J ( B ) are. respectiuely, the rank cornplements of -4 and B in

(1,) The matrices .i9 B, and the two terms G1 = Jf(-4) a71d G2 = . l t(B) in the rigf~t-f~rrnd side of

Eq.( 11.19) satisfy the following several equalities

Proof. The ecluivdence of Eqs.(11.17) and (1.18) is derived from Eq.(l. 1 3). We know from Tlicoreni

9.8 thüt under the condition (1 1.17), the Moore-Penrose inverse of 1 can be esprrrrod as

Tlien put ting it in Ecl.(ll. 15) imrnediateiy yielck

Tlieorem 9.9. 0

Eq.(11.19). The results in Part (b ) are derivecf froni

Theorem 11.5 (Tian, 1998a). Let -4 + il3 be an m x 71 contplex mat&, wfaere -4 and B ure two r d

matrices. und suppose that A and B satisfiJ

or. erpivalently

R(.4) C R(=i i iB) und R(-4') C R(-4T * i ~ ~ ) . ( 11-21)

Tfien the Moore- Penrose inuerse of -4 + iB c m be expressed us

t ( -4 + LB)' = Gi - iG? = [EB,(.4 + B ; L + B ) F ~ , 1' - i [ ~ 5 ; ( ~ ( B + ~B'..I)F..I, ] . (1 1-22)

iuhere -4' = EB.4, -4.' = .4Fs, BI = E-4B and B:! = BFA.

Proof. Fo1lou.s directly from Tbeoreni 11.4. 0

Corollary 11.6 (Tian. 199Sa). Suppose that -4 + iB is cr nonsingular complez rnrrtrilr. whem -4 rrritl B

(LW rcd.

( a ) If 60th -4 and B are nonsingdar, then

( -4 + iB )-' = ( -4 + B-I-' B )-' - i( B + - 4 ~ ~ l - 4 )-' -

(11) If 60th R ( 4 n R(B) = { O ) and R(A*) n R(B*) = { O ) , then

( -4 + iB)-' = ( E B - - ! F ~ ) + - z(E.-IBF.~) ' .

( c ) Let -4 = XI,,, , wliere X zs a real number such that XI,, + iB is ~tonsinyular. t h e n

(Xi , , +iB)- ' = X(X21, + B 2 ) - ' - ~ ( x ' B + B ~ B " ~ ) ~ .

Proof. Folloms clirectly from Theorem 11 -4. CI

Ive n e ~ t turn Our attention to the ?doore-Penrose inverse of the sum of k matrices. and give sorne

gcrieral formulas.

Theorem 11.7 (Tian, 1998a). Let --II, - - - . ,Ar; E C n ' x n be given. If t tmj sutisfy the follou;iny r fmk

arlditiuity condition

r( -4) = kr[ -Ak Y - - - : -Ak ] = kr[.Ai. - - - , -4; 1: (11.23)

iuttere -4 is the cil-culant block rnntrü defined in Eq.(ll.6)? then

(a) The Moore-Penrose inverse of the sum XI=, ;li cari Iie expressed us

ruf~w-e J(--Li) is the rank complernent of ,4;(1 < 2' 5 k ) i n -4.

(b) The runk of J(--Ii) is

zuftere i 5 i < k. D, ia the ( k - 1) x ( k - 1 ) block matrix rcsulting j?om the dcletion of the first block ro~u

( ~ n d itlt hlock co1.i~rn.r~ of -4.

(c) =I l , -A2, - - - . .Aa and .J~ (--Il), .lt(,.l?), - - - , ,1+(-4~) satisfg the following Lwo cquakities

Proof. Follows from the combination of Theorem 9.18 with the ecluality in (1 1.13).

Corollary 11.8 (Tim, 1998a). Let -4.': - - - , .-ln: E Cu'"". If tfteg sntisfv t f ~ e followirtg rnirh- rldrfitivitg

con dit ion

wher-e a; arrd ,fii ur-e

Proof- We first show that under the condition (11.17) the rank of the circulant matrix -4 in Eq.(LL.G) iç

According to Eq.(11.7), we see that

C'rider Eq.(11.26)1 the r a d s of al1 .Ji are the same. that is,

This n-e have Eq.(11.28). In that case. applying the result in Corollary 9.19 to the circdant block rnacris

-4 in Eq.(11.13) produces the equality (11.27). O

At the end of this chapter, we should point out that the formu1a.s on Moore-Penrose inverses of sums

of matrices given in this chapter and those on Moore-Penrose inverses of block niatrices given in Chaptcr

9 are. in fact. a group of dual results. That is t o say. not only can -ive derivc 1Ioore-Penrose irivtmes

of sunis of matrices frorn Moore-Penrose inverses of block matrices. but also n-c can mrike a coritrar:-

(lerivation. For simplicity: here we only illustrate this cassertio~i by a 2 x 2 block niatris. In k t . for ariy

2 x 2 block matrix can factor as

If 111 satisfies the rank aclditivity condition (9.22), then !VI and % satisfi;

Hence by Theorem 11.4, Ive have

wiiere .J(Nl) aid I ( & ) are, rerpectively. the rank complesieiiti of Nl anri & iii 1 cl z: 1 . \I-ritt.m

L A

in ari esplicit forni. Ec1.(11.29) is esactly the formula (9.33).

In addition. we shalI menti011 another interesting fact that Eq.(lI.lG) m r r bt. (xtcnciecl to ariy rcal

qi~aterriion matris of the form '4 = -Ao + i-ql f j--12 + k.-A3, where --lo-.-lAi are real m x T L matrices arid - .'l b - = J - = k'> - - -1, i j k = -1, a s Follows:

Their proofs will be given in the author's paper [97]. Furtherrnore. this work c m even L>e estended to

matrices over any 2n-dirnensional red Clifford algebras through a set of universal sirnilarity factorization

eqidities estabtished in the authar's recent paper f96].

Chapter 12

Rank equalit ies Moore-Penrose

Let

for submatrices in inverses

I x ;L 2 x 2 block matris over C. where -4 E Cm" ". B E Cln x k . C E Ci"" D E C l X " . aricl let

M o r e o ~ e r ~ partition the Sloore-Penrose inverse of A4 as

wliere Gr E CnX"'. -4s is well knoivn, the expressions of the subrnatrices GI-G.: i ~ i Eq.(12.3) arc quite

coniplicated if there ,are no restrictions on the blocIcs in s\.l (see, e-g.. Hurig and Markhani. 1975, hliao.

1900). In that case, it is hart1 to find properties of submatrices in Agt. In the preserit cbapter: ive corlsidcr

;L sirnpler problern-what is the rankç of subrnatrices in M t ? when ALI is arbitrarily given? This probler~i

KU examinecl by Robinson (1987) and Tian (1992~) . In tbis chapter, n e shall gïve this probIcrri a new

discussion.

Theorem 12.1. Let RI a n d M t be giuen by Eq.(12.1) und(12.3). Then

where Pi, 14, T4,7i nnd I.VL are defined in Eq.(12.2).

Proof. We only show the first equality in Eq.(12.4). In fact Gl in Eq.(12.3) can bc written as

Therr applying Eq.(2.1) to it we find

establishing the first equdity in Eq.(12.4).

Coroilary 12.2. Let and M t be given by Eqs.(12.1) and (12.3). If

r ( M ) = r (V l ) + r ( & ) , i.e., R(VL) n R(I:') = {O},

~ J L C T L

4 -4-B r ( G 3 ) = r 1 " " 1 - r [ 1 j 17.9)

C'C C'D

ProoF. Urider Eq.(12.7), we also know that R(Vl) n R(V2D* PI/:I) = { O ) . Tlius the first eqiiality in

Eq. ( 12.4) becomes

estahLishing the first one in Eq.(i2.S). Sirnilarly, we can show the 0 t h three in Eqs.(l'2.S) arirl f 12.9).

El

Similarly, we have t h e foIlowing.

CoroIlary 12.3. Let AJ and M t be given by Eqs.(l2.1) and (12.3). If

( 1 ) = T V + 1 ) i e R(CVr;) n R(Wi) = { O ) ,

4 BD' 1 - r[C, O]: r(G2) = r [ : ] - r[ -4: B 1. (LZ.11) C DD*

-TIC, DI, r (G4) = r [ " " 1 -r[-4. BI- ( 12.12) CC' D C-4' D

Conibining the above two corollaries, we obtain the following, which is previously show-n in Corollar-y

9.9.

Proof. We only show Eq.(l2.14). Under Eq.(12.13): we find ttiat

r(V2DLPt:>) = r BD'C B D ' D 1 = r ( D ) . DD'C DD'D

Tllus the first equality in Eq.(l2.4) reduces to Eq.(12.14). O

Corollary 12.5. Let Ar( crnd M t be given b y Eqs.(12.1) und (12.3). If AI strtisfies the ~mnk addi t iu i t !~

con dit iu n

r ( M ) = r(-4) + r ( B ) + r (C) + r ( D ) . (1TlS)

Proof. Follows ciirectly Frorn Eqs.(l2.14)-(12.17). a

Corollary 12.6. Let M and iW bc given b y Eqs.(lLl) und (12.3). If

r(G3) = r

Proof. The inclusion in Eq.(12.20) irnplies that

Tlius the two rank equdities in Eq.(Z2.4) become

xitl clic two rank ecludities in Eq.(l2.5) become

Hericc we have Eqs.(l2.21) and (12.22). 0

S inlilarly, me Iiave the following.

Corollary 12.7. Let 1\/1 and h o be given b y Eqs.(22.1) und (22.3). If

Conibining the above two corollaries, we obtain the followirig.

2 10

Corollary 12.8. Let dl und kit be giuen by Eqs-( 12.1) urrd ( 1 2 . 3 ) - If

r ( M ) = 4 - 4 1 .

then

( G ) = ( 4 ) r (G2) = r ( C ) , r ( G : s ) = r ( B ) -

n 71 cl

Proof. Clearly ECI-(12.26) iniplies that r(iC1) = r(\,-; ) = r ( L l Y l ). TIius we have Eq.( 12.27) i)y Coroili~ricts

1Z.G aicl 1'2.7. On the otIier liand. Eq.(12 .26) is also ecluib-dent to -L-ltB = B. C.-lt-4 = C and D = C.4'B

by Eq.( l .5) . Hence

which is Eq.(12.28). 0

Sext me list a group of rank inequalities derived from Eqs.(l2.4) and (12.5).

Corollary 12.9. Let M a n d M t bcgiven by E q ~ ~ ( l 2 . 1 ) rrnd ( 1 2 . 2 ) . Then the ru7~X: of G , i7~ .snti.s$e.s

Proof. Observe that

Putting them in the first rank equality in Eq.(12.4), we obtain Eqs.(12.29) aiid (12.30). To show

Eq.(l2.31), we need the folloming rank equality

whicfi is derived from Eq.(1.6). Norv applying Eq.(12.32) to PMtQ in Eq.(l?.G). we obtairi

Rank ineclualities for the block entries G2, Gq ;inci in Eq.(12.3) can also bc tlerived in the siniilar

W ~ J - shown above. Finally let D = O in Eq.( lS. l ) . Theri the results in Etp~( l2 .4) and (12.5) can tir

sirnplified to the following.

Thoerem 12.10. Let

A& =

n d denote khe Moore- Penrose inverse of LW US

Various consequences of Eqs.(12.35) ancl (12.36) can also be derived. But we omit thcrn herc.

Chapter 13

Ranks of matrix ex~ressions t hat involve Drazin invekes

As one of the important types of generalized inverses of matrices, the Drazin irivcrscs aricl thcir applicatiuris

have well been examined in the literature. Having established so many rank ecpalities in the prccc!tliri::

chapters. one niight naturally consider Iiow to estend ttiat work frorn lloore-Perirose inverses to D r u i n

inverses. To clo tliis, we only need to use a basic identity on the Drazin inverse of a rnatris = L ~ = -4k( -42k~i ) i -qk (sce. e-g., Campbell and Meyer. 1979). In that cise. the rank formulas obtaincci i r i r . h

prccccling chaptcrs can al1 be appIied to establish various rarik equalitics for niritris espressions involving

Draziri inverses of matrices.

Theorem 13.1. Let -4 E C V L X m with h d ( - 4 ) = k . Shen

( a ) r( I,,, + --ID ) = r( -Ak+' + -4' ) - r(.-lk) + m.

(b) r ( In , - ( . - L D ) ' > 1 = r ( + .-Ik ) + r( . A k f L - =Ik ) - 27-(*49 + ru.

Similarly we <:an find r( In, +.-ID ) = r( .4ktL +- A" ) - 4 - 4 9 + m. Note by Ec1.(1.12) tliat

Thus Par t (b) follows frorn Part (a). O

Theorem 13.2. Let -4 E C n L X m with Inci(-4) = f i . Then

( a ) r ( -4 - -4-4* ) = r ( -4 - -AD-4 ) = r( = L P C 1 - --Lk ) + r(=L)

(b) r ( -4 - -4-4 -4 ) = r(-4) - r ( - ~ ~ ) -

( c ) =lAD = . - IDA = -4 e A2 = -4.

( c i ) - 4 . 1 ~ ~ . 4 = -4 Ind(A) 5 I.

The results in Theorern 13.2(d) is well known, see, e.g., Campbell ancl Meyer(1979).

Proof. Applÿing Eq.{1.6) to r( -4 - -4-.1~ ) yields

lis recpircd for Part (a). Sotice that --ID is an outer inverse of -4. Thus it follow by Eq.(.5.G) tha t

as rcquirecl for Part (b). The results in Parts (c) and (cl) follow frorn Parts (a) arid (b ) . n

Theorem 13.3. Let -4 E C"'""' with Ind(-4) = k.

(a) r ( -4 - --lD ) = r( --lk+' - -4k ) + r (-4) - r(AL)-

(b) r ( - 4 - - - lD) = r(A) - r(--ID). Le.. =lD sr, -4 .A"+' = --lk.

( c ) -4D = -4 -A3 = -4.

Tlie results in Tlieorem 13.3(d) is well known.

as required for Part(a). The results in Parts (b) (c) foIlow inimediatcly from ic.

Si~riilarly, we can estüblish the following two.

Theorem 13.4. Let -4 E C"'"" wiYt Incl(..l) = k.

(a) If k >. 2. then r( -4' - = L ~ ) = r( -4kf" .qk ) + .(-A2) - r( . - lk) .

(b) If k = 2, then r(i12 ---ID) = r ( A 5 - A2).

(c) Ifk = 1, then r ( ~ ' - .rlD) = T(-4.' - -4).

(d) -4" =.4D a -4" = A when k = 1.

(e) -4' = --ID a -4" -4' when k = 2.

Tlic two equivalence relations in Theorern 13.4(6) and (e) were obtciinecf by Grass and Trenkler(1997)

when they considered generalized and hypergeneralized projectors.

Theorem 13.5. Let -4 E C m X m with hd(-4) = k.

(il) If k 2 3. then r( .d3 - = I ~ ) = r( -4k+4 - -4' ) + ~ ( - 4 ~ ) - r(,-lk ) .

( b ) If k = 3. then r ( A 3 - - - I D ) = r(-q7 - & d 3 ) .

( c ) If k = 2, then r( -4" -qD ) = r(-46 - -q2).

( c l ) I f k = 1. then r(-4" -.-ID) = r(--I"--i).

( c ) ,-13 = ,4D a -4' = -4" when k = 3.

( f ) -4" -4D -A6 = -4' when k = 2,

( g ) -A3 = ,-lD ,-I" = -4 when k = 1.

-1 square matris -4 is said CO be quasi -idempotent if -4"' = -4" for some positive integer k. Iri a

rccetrt paper by Mitra, 1996, quasi-idempotent matrices and the related topics are weI1 esaniincci. The

rcsults given beIow reveals a new aspect on quasi-idempotent matrices-

Theorem 13.6. Let -4 € C m "" with Ind(A) = k. Then

(a) 1 - [ - 4 ~ k (--ID)'] = r( =LMf' & -4' ).

( b ) (-4")' = --ID, i.e., =lD is idempotent -4"' = -4" ie., -4 is q~rasi-iderripotent.

Proof. B y Eq.(2-3) anci ( . A ~ ) ' = (J42)D we find that

Sirnilarly. we cari obtain r[ .4D + (=lD)' ] = r( .4G1 + -4' ). The result in Part (b) follon-s irnnieciiiltely

frorn Part (a).

-1 square matris -4 is said to be quasi-idempotent if -4"l = .-t"or sorne positive integer k. Ir1 ;L

recent paper by Mitra(1996), quasi-idempotent matrices and the relateci topics twre well csi~rrii~icci. Tlitt

rcsults given below reveal a new 'aspect on quasi-idempotent matrices.

Theorem 13.7. Let A E PX" tuith Ind(A) = k . Then

( a ) i[.-lD - ] = r( AkCL + -4k ) + r( i l t+l - --Ik ) - 7*(.4k).

(b) The following three statements are equivalent:

(1) ( A ~ ) ~ = -4D, ive., AD is tripotent.

(2) r ( -~" .~ + A k ) +r ( .4* iL - -4') = r ( ~ * ) .

(3) R( -A'+' + -4' ), R( -4"' - .4' ) = { O ) and R[( -4"' + -4' R(( .A*-' - ri")'] = { O ) .

Proof. Applying Eq.(1.14) and Theorem 6.3(a) to .4D - (-dD):' yields

ris required for Part (a). The equivalence in Par t (b) follows directly from (a). 0

Theorern 13.8. Let -4 E Cm" with Ind(,4) = k. Then

(a) r[ -4--ID - ( ; ~ - 4 ~ ) * ] = 2r[d\ (-4")' ] - 2 r ( ~ " ) .

(b) .4=iD = (-4--1~)' R(-L*) = ~[(.-l~)*] i e . . -4' is EP.

Proof. Xote that both .4-qD and (-4.qD)' are idempotent. It follows from Eq.(3.1) ttiiit

;LS required for Part (a) . The resuIt in Par t (b) follows imrnediately froni Part ( i t ) .

Theorern 13.9. Let ,-l E Cr"X"' . with Ind(,4) = k. Then

(Ii) r(--Lt - A4D) = r(..Lt) - r ( ~ ~ ) , i.e.: ;lD sr, ..LI o R(;tk) C R(.4") and ~ [ ( - 4 " ) - j C

r(.-l), i.e., -4 is povuer - EP.

(c) I/ Ind (-4) = 1, t f ~ e n r( .At - ) = 2r[ -4, -4' ] - 2r (-4).

Proof. Sincc botti .4t and -4D are outer inverses of -4. it folIows frorn Eq.(5.1) that

as recliiirecl for Part (a). The results in Par t (b)-(d) Follows imrnediately frorr, Part (a). a

Theorem 13.10. Let -4 E CnLXm with Ind(-4) = Ic. Then

-4'; (e) r [ ~ * ( i l ' ) ~ - - 4 ~ ~ 1 = r - r(-4")- I (-4*)* l

L -L

( f ) r[ (.4")t,lk - .qD--l ] = r [ d \ ((A*)" - r ( ~ ' ) .

Proof. Note tha t both -4--lt and --l--ID are idempotent. Then it follows by Eq.(3.1) tliat

as required for Part (a). Similarly we can get (b). The result in Part (c) folloms imrnediatcly frorn Parts

(a) and (b). sirnilarly we cari show Parts (e)-(g). O

Tlieorern 13.11. Let -4 E Cr""" witfr Ind(A) = k. Tften

(b) .4idD = .-LD.4t a R(-49 ) R(A*) und ~[(.l")'] C R(.4) i.e.. -4 is poiuer - EP - .4'.4"

-4..-LT.

Tlie second one in Part (a) follows from Theorem 6.4. The result in Part (b) follows imrnediately from

Part (a). (7

Theorem 13.12. Let il € CmX" with Ind(A) = k. Then

4 " ( 4 - - 4 4 ) 4 O *Ak -4-

(a) ~ ( A * . - L ~ - ; ~ ~ . 4 * ) = T

Proof. Applying Eq(2.3) to ,4'.,lD - -4D-4L yields

as recpirect for Part (a). Parts(b) and ( c ) follow from Part (a). Next applying Lemma L.'L(f) to tiic rarik

ccy.dity in Part ( a ) yields Part (cl). O

Theorem 13.13. Let -4 E C n Z X m with Ind(.4) = k. Then

4 4 (Ak)*

(Ak) *

(11) r-[ -4 - (-4 ] = ~ ( ~ 4 ) - r(.4*), zf -4" zs EP-

(c) (-4D)t = -4. -4 is EP.

Proof. According to Cline's identity ( L L ~ ) ~ = (.4k)t=l'"L(~k)t(see [IO] and [22J). wc find liy Eci.(2.S)

t lint

as recpired for P,wt (a). The results in Part (b) and (c) follows imniediateiy from Part (a). [7

Theorem 13.14. Let -4 E CmX"' luith Inci(.-1) = k. 27ien

Proof. It is easy to verify that both - 4 ~ ~ r l and (,4D)t are outer inverses of --ID. In chat case i t follows

frorn Ec1.(5.1) t h

as requirerl for Part (a). The result in Part (b) folIows immediate~ÿ from Part (a). n

Theorem 13.15. Let -4, B E Cm"" with Lnd(A) = k und Ind(B) = 1. Then

(a) r ( - 4 , ~ ~ - B B ~ ) = r 1 1 + r [ .Ak; B1] -.(.A.) - r ( B L ) .

(b) -4-4" = B B ~ o R('4') = R(B1) and R[(iIk)-] = R[(B')']. - - ( c ) r( -4--ID - B B ~ ) is nonsingular r = r[ -4": Bi ] = ~ ( - 4 " + r-(B1) = I I I

R(-4") G R(B1) = R[(- - lk)*] e R[(B1)*] = Cm.

Proof. Note that both LA^ and BBD are idempotent. Then it follows from Eq.(3.1) that

as recpired for Part (a). The results in Parts (b) and (c) follow imrnediateiy from Part (a). C

T h e o r e m 13.16. Let -4, B E C n L X r n with Lnd(,4) = k. Tiren

( c ) ~ - 4 ~ B = l?AD-4 CJ R(BA*') = R(-4k) und R((-4' B) ' ] = R[(tlk) '] .

Proof. Note =1-4D = -4D~4 is idempotent. Tt follows by Eq.(4.l) tlirit

Tlius ive have Parts (a). Replacing B by .4t in Part (a) aricl siniplif).ing it yicltls the t in t c.qiialir~- i r i Parr

(t)}. The second ecluality in Part (b) follows from Theorern 13.5(a). 0

Theorem 13.17. Let -4 E B E Cm""' with Ind(B) = k nnd C E C n X " iuitlr Inci(C) = 1 . Ther~

Proof. Note that both BB. and C ~ C are idempotent. Ttien it follows from Eq.(4.l) tlint

as required for Part (a). 17

Theorem 13.18. L e t -4 E C m x n , B E C''lx with I n d ( B ) = k- nrrd C E CTLX" 1~IZ'th Ind(C) = 1 . Then

Proof. Xpplying Eq.(l.ï) to A - BB~.-I yields

as recpired for Part (a). Siniilarly we c m show Parts (b) and (c). 17

Theorem 13.19. Let -4, B E C n I x m with Ind(,-l) = k nnd Incl(B) = 1. Then F Y

-42k -4k@ ( a ) r ( -4B - - 4 ~ ~ ~ = 1 ~ - 4 ~ ) = 7- 1 Bt-Ik i - d-4") - i (Bil-

-42k -Ak Bi (b) ~ ~ - 4 ~ € { ( - A B ) - ) r 1 @-dk ~ ' f 1 = r( , - ik) + r ( B L ) - r ( -AB) .

Proof. It follows by Eq.(2.9) that

Thus we have Parts (a) and (b). 0

Theorem 13.20. Let -4. B E C m X m with Fnd(-4) = k and Ind(B) = 1. Then

ProoL Xote chat botli .-L-iD = AD.4 and BBD = B D ~ are idempotent. Tlien it follows bu Eq.!J..>û)

t h t

iw rc?c~uirecl for Part (a). O

Theorem 13.21. Let -4, B E C"nX"L with Ind( -4 + B ) = k a71d deno te :Y = -4 + B. Tfwn

(a) r(.-1ND B ) = r(-4.~i" + T.(N*.B) - r(IVb).

((1) . 4 i v D ~ = f 3 ~ ~ - 4 o R(B!V~ )C R ( N ~ ) and R [ ( N k B ) * ] C R[(N') ' ] .

Proof. It foIlorvs by Eq.(l.ï), tha t

which is the first equality in Part (a). Tlie second equality in Parr ( a ) Follows from r(-iAr" ) = r( -4.v LI ) . i - ( :\'" B ) =

r-(ivD~) and r ( x D ) = r ( ~ ' ) . Unclcr R(B) C R(N*) and R(rLs) C R[(N')*). i t follorvs rliat r ( .4Sk) =

r(-4) and r ( ~ ' B ) = r ( B ) . Thus Part (a) becomes Part (b). Nest applying Eq.(2.3) to - 4 ~ ~ B - L?:\rD-4

Thus we have Parts ( c ) m d (ci). D

Theorem 13.22. Let -4. B E CmX"' be giuen, und let LV = -4 + B witfl Ind( -4 + B ) = k. Then

(W

Proof.

Ind( -4 + B ) 5 1 und r( -4 + B ) = 4.4) + r ( B ) .

It follows by Eq.(l.T) that

w-hich is exactty Part (a). Note that r(iv9 ) r (N ) = T ( -4+ B ) 5 r(-4) +r(B) . Thw r(1Vk) = r(-4) +l-(B)

is eqiiivalent to Ind(N) 5 1 and r ( N ) = r(-4) -t r(B). CI

111 general we have the following.

Theorem 13.23. Let AL, -A2, - - - . -Ak E CnLXm lUith Ind(iV) = k. where N = --Li + --L2 t + + + -Ak. und

denote -4 = [hg( -41 , -q2, - - - , ) . Then

Proof. Since .Ai are square, Eq.(11.7) can be written as

Tri that c'ase, it is easy to verify tha t

; ~ r i c I

[diag( Ji: .&, . - - , J k ) j D = diag( J?. .J.f. - - - . . I F ) This LW have

which is Eq.(13.1). [7

Chapter 14

Rank O equalities for submatrices in Drazin inverses

Let

be a square block matris over C, where -4 E C r n X m anci D E CnXn.

and partition the Drazin inverse of :'LI as

wfiere Gl E C n t X n L . It is, in general, quite difficult to give the expression of Gl -G., . In this c1iaptc.r ive

consicier a sinipler problem-the ranks of the submatrices Gr-G., in Eq.(14.3).

Theorem 14.1. Let AI und MD be given by Eqs.(14.1) and Eq.(14.3) w i t f ~ Inc l (M) 2 1. Then the r m b

O! Gr -G.{ i.n Eq.(14.3) cars be deterrnined by the followiny fo7-rnvla.s

iuhere L i , I.2. I.t; and PV2 are dejïned in Eq.(14.2), und

Proof. We only show Eq-(14.4). in fact Gl in Eq.(14.3) can be written as

where Pl = [I,, 01 and QI = . Then it follows by Eq.(1.6) and block elementary operations

t ha t

which is exactly the equality (14.4). n

The further simplification of Eqs.(14.4)-(14.7) are quite difficult. because the powers of J I appcar

in tliern. Howevcr if r l l in Eq.(l4.1) satisfies some aclditiorial conditions. tlic four rank eqtrditics i r i

Eqs.(l4.4)-(14.7) can reduce to sinipler forms. We next present sonie of them. The first one is r c h e c i

to the n-ell-known result on the Drazin inverse of an upper triangular blocli matris (see Campbell and

hlcyer. 2979).

Theorem 14.2. T f ~ e rank of the subrnatrïx ,Y in Eq.(l4.9) is

In the same manner we can show the other three in Eqs.(l4.13) and (14.14). 0

Corollary 14.4. Let ibf be yîven by Eq.(14.1) &th Ind(h1) = 1.

( a ) If l'LI satisfies the rank additivity condition

r ( M ) = r ( V I ) + r(V2) = r(CL;) + r ( 1 l ) .

then the ranks of G L -GA! in the group inverse of r t l in Eq.(14.3) can be expressed us

r (Gi) = r ( V l ) + r(WL) + r(V2DFYz) - r(lb1).

r ( G 2 ) = r (V2) + r(FVl) + ~ ( k i BW2) - r ( M ) .

r (G3) = r ( l 2 ) + r(LiTL) -+ r(\GCI,i-L) - r ( J 1 ) .

r(G.,) = r (L) + r(FV2) +- r ( b i --UVL) - r ( M ) .

( b ) If Af satisfies the rank additivity condition

r ( M ) = r ( A ) + r ( B ) + r (C) i- r ( D ) .

then the ranks of G L -Gai in the group inverse of ib1 in Eq.(14.3) sirtisf?J

( G ) = ( 4 - r ) + r D L ) r ( G ) = r ( B ) - r ( C ) + r ( I ; Br[:>).

r ( G 3 ) = r ( C ) - r ( B ) + r(V:>CLVi), r (G2) = r ( D ) - 4-4) -+ r ( l i --K)-

whe7.e b i , V,, I.V1 und Pb are defined in Eq-(14.3).

In acldition, we have some inequalities on ranks of submatrices in the group inverse of a block rnatris.

Corollary 14.5. Let 111 be given by Eq.(14.1) with Ind(l\f) = 1. T h e n the runks of the mutrices Gr 4.1

i r ~ (14.3) .sm%jg the following rank inequalities

( i i ) r ( G L ) 2 r ( l ) + r(Li.1) - r ( M ) -

(b) r ( G l ) 5 r (1 ; ) i I - ( C C - 1 ) + r ( D ) - r ( M ) .

( c ) r ( G 4 > r(C:> j + r(LV1) - r ( M ) .

[ci) r(G2) 5 r(Vb) + ~ ( W L ) + r ( i 3 ) - r ( M ) .

( e ) r(G3) 2 r ( V l ) + r(l)lL2) - r ( M ) .

( f ) r ( G 3 ) 5 r ( L i ) + r(CV2) + r ( C ) - r ( M ) .

( g ) r ( G 4 ) 2 r(L5) + r(CV2) - r ( M ) .

(11) !* (Gd) 5 r (b>) + r(CV2) + r(.-i) - r(iL1).

Proof. Folloivs from Eqs.(14.13) and (14.14). O.

Chapter 15

Reverse order laws for Drazin inverses of products of matrices

In this chapter we consider reverse order laws for Drazin inverses of proclucts of matrices. C\'e will

giw necessary and suficient conditions for ( = L B C ) ~ = cDBD--ID to hold and ttien present sorne of its

c:orisecluences.

Lemma 15-1. Let -4, S E with Ind(A) = k. Then S = AD if and only if

Proof. Follows froni the definition of the Drazin inverse of a niatris. 0

Lemma 15.2. Let -4. B. C E C'"Xn' wath Ind(-4) = k L t Ind(B) = k2 nnci Inci(C) = k:$. Then the

prarluct cD ~ ~ - 4 ~ of the Dwzin inverses of A: B. und C cart l e q r e s s e d in the / o r m

tuhere P. LV I L ~ L ~ Q sutisfy the three properties

Proof. It is easy to verify that the 3 x 3 block rnatrk X in Ecl.(l5.2) satisfies the conditions in Lemma

S.S. Hence it folIows by Eq.(S.S) that

( ~ 2 h ~ + 1 ) t ~ k j ~ k = ( ~ ' > k ~ + L ) t ~ k ~ - ~ k ~ ( ) * *

:vt = * * * ( 15.4)

* O O

Tliirs w e have Eq.(l5.2). The three properties in Eq.(15.3) follows from the structure of Ar. 0

The main results of the chapter are the following two.

Theorem 15.3. Let -4, B, C E CnLxm with Ind(A) = kLt Ind(B) = k2 and Ind(C) = k:!: i r n d dcnote

A l = -4BC with Ind(M) = t . Then the reverse order Law ABC)^ = C D B D ~ D /LOI& if ~ n d only if -4. B

( L T L ~ C satisfg the three rank equalities

Proof. Let S = c ~ B ~ = L ~ . Shen by definition of the Drazin inverse. S = d l D if arid orily if

J I t f '-Y = ML, .\L'dlLf' = l\It and r(x) = r(h.f t) , which are equivalent to

Replâcing S in Ecl.(l5.S) by .Y = pNtQ in Eq.(13.2) and applying Eq.(l.i) t h r i . w t t fincf chat

Piitting hem in Eq.(15.S), WC obtain Eqs.(l5.5)-(15.7). (7

Theorem 15.4. Let -4. B. C E Cm""' with Ind(,-l) = k i : Ind(B) = k2 and Intl(C) = k:<. and let

JI = .ABC witl~ Irid(A1) = t . Tf1e.r~ the reverse order Law ( . - I B C ) ~ = c ~ B ~ - ~ ~ f~olrls if ( m d 0 7 1 1 ~ if -4. B

c ~ n r i C sutisfg the followiny rnnk equality

T h s Eq.(la.S) follows by putting P, 1V and Q in it. 0

\Ire next give some particular cases of the above two theorems.

Corollary 15.5. Let -4, B. C E Cm""' with Ind(B) = k and Id(-ABC) = t . tul~ere -4 and C arc

(a) r[(--WC) - C-'B*=~-' ] = r + r[ Bk7 C(-4BC)'I - r ( B 9 - r[(=LBC)L].

(b) (--WC)* = C-L BD-4-L e R[C(.4BC)L] =R(B' ) and R{[(.-iBC)t.4]') = R[(BVm].

Proof. i t is easy to verify that both (,ABC) and ~ ~ - 4 - l are oiiter inverses of ,-1BC. Tliiis it

follows from Eq.(5.1) that

as requirecl for Part (a). Notice that

Then Part (b) follows from part (a). CI

CorolIary 15.6. Let A, B, C E C n L X r n uiitlr Irid(,4) = Ici, Ind(B) = k:! and Iritl(C) = k;$. m ~ d let

31 = -4BC with Incl(M) = t. Moreouer suppose that

Tfreri the reverse order Law (..IBc)~ = cD13 D.4D holds if and oniy if .A. B und C sotisfg Eq.(15.7).

Proof- It is not difficult to verify that under Eq.(15.10), the two rank equalities in E.qs.( 15.5) ;mcl ( 15.6)

become two identities. Thus, Eq.(l5.7) becomes a necessary and sufficierlt condition for ( + ~ B C ) ~ =

C ~ B ~ A ~ to h ~ l d .

Corollary 15.7. Let -4, B E C m x r n with Ind(d) = k. L.id(B) = 1 and Ind(.4B) = t . Then the follou~ing

Proof. Letting C = I,, in Eq.(l5.9) results in Part (b): and tetting B = 1,,, arid replacirig C by B iri

Theorern 15.4 result in Part (c). O

three are equiualent:

(a) (-4sjD = ~ ~ - 4 ~ .

I -

0 -42k+1 -4k O

~ " + l ~1-4" O 0 (11) 7-

BL O O (-4 B)

O 0 4 ) (,4B)'t+1

= r(--Ik) +- r ( ~ ' ) - r[(. .IB)L].

( c ) Tite three runk cqvalities are ail scrtisjïed

Chapter 16

Ranks equalit ies related to weighted Moore-Penrose inverses

Ttie weighted hloore-Penrose inverse of a niatrix -4 E CnlX * with respect to two positive definite matrices

JI E C m x m ancl AT E C n X n is ctefined to be the unique solution of the following four mat rk equations

aiid tliis -Y is often denoted by .Y = .4:f-Ar. In particular. when = In, ;uid -V = In. .-L\~.-, is

the standard Moore-Penrose inverse ,-lt of -4. -1s is well known (see. e-g.. Raa ;incl 1Iitra. I911). clic

wiglitecl bloorcPenrose inverse -4:,*,, of -4 can be written as a marrix espressions involvirig n standard

1 bore-Penrose inverse as follows

wliere ilfi and 11-f are the positive definite square roots of 1l.i and N. respectively. According ro Eq.(16.2).

i t is easy CO verif$- that

Bnsetl on tliese basic facts and the rank forniulas in Chapters 2 -5 . u-e riow (-:in cstablisli varioris

ri~rik equalities related to weighted Moore-Pe~irose inverses of matrices. arid the corisitlcr clicir \;trious

cowiequences.

Theorem 16.1. Let -4 E C m X n l e yiven, hl E C r r L X n L and iV C n X r z be twu positiue definite nmtrices.

t (a) r(-4t - -4iLf.Lv) = r

( c l ) .4',,, = .-Il e R(iLf-4) = R(-4) and R[(..IN)'] = R(.4-).

Proof. Note that .4t and -4LVN are outer inverses of -4. Thus it folloivs froiri Eq.(Ei.I) that

Parts (a)--(c) follow imrnediately from it. O

Theorem 16.2- Let -4 f C m x n be giuen, 1l.d E C m X "' und 1V E C n X n bc two positive definite mutrices.

Th en

(a) r ( -4-4Lf.1,, - -4--Lt ) = r[ -4. M.4 ] - r(-4).

(c) ;~4: , , , = -4,lt R(M.4) = R(A).

Proof. Xotc that both -4.4t and -i l ir ,Iv are idempotent. It follonrs From Eq.(3.1) that

as rccluired for Part (a). Similarly we can show Part (b). f7

Theorem 16.3. Let -4 E C r n x m be given, und N f Cm"" be two positive definite mutrices. Then

t (a) r - 4 4 ) = r 4 , M.4 1 + r [ -Ae, 1V-41 - %-(-A).

(b) .4.4:,., = -4: ,.,. 4 u R(M.4) = R(N-4) = R(;La) both MA und iV.4 W C EP.

Proof. .\:ore that both -4.4t and -4.4ir.iv are idempotent. It follows by Eq.(3.1) thnt

-&, A7

= T- [ ] + r[ -4. .4LrTlV ] - 2r(--i)

= r [ 'y:'* ] + r[--L. N-' . - l*] - 'i!7-(.-l).

as required for Part (a). Part(b) follows immediately from Part (a). 0

B'ased on the result in Theorem 16.3(b), we can extend the concept of EP rnatris to weiglited case:

A square matris -4 is said to be weighted EP if both Ad-4 and N-4 are EP. where both 1\ l arid 3' are two

positive definite matrices. I t is espected that weighted EP niatrL~ worrtd have some riice properties. But

ive do not intend t o go further dong this direction in the thesis.

Theorem 16.4. Let -4 E C m X m be giuen, and M , 1V E C m X n L be two positiue definite matrices. Then

(a ) r( - u L f e i V - ; L M , ~ A ) = r[--LT, M.41 + r [ i l T . ATT-41 - 2r(-4)-

Proof. Follows from Eq.(3.1) by noting that both i l 4 O f . , and A4;,, are idenipotent. 0

Theorem 16.5. Let -4 E CnlXn' je giuen uiitfr Ind(.-l) = 1, crnd M, N E PX " 6e twa positive riefinite

rrmtrices. Then

( a ) r ( . - ~ \ ~ . ~ - . A X ) = r [ A œ 7 Lb1-41 + r [ A * , ~v.41 - 2r(.4).

(II) AL,.,, = --L# u R(M.4) = R(N.4) = R ( A X ) , i-e., .4 is weighted EP.

Proof. Sote that both --lt and -4" are outer inverses of -4. It folloas by Eq.(5.l) tliat

as recpired for Part (a). 0

Theorem 16.6. Let -4 E Cmxm je giuen with Ind(A) = 1, and M. :\' E C"l'X"' be two po.sitive tlcfinitc

rrrrttï-ices. Then

Proof. Sotc ttiat both -4-4i and -Ail# are idempotent. It foHoms from Eq.(5.1) tliut

as required for Part (a). U

Theorern 16.7. Let -4 E CnLXm be given with Ind(-4) = k . and AIt iV € CmX"' be two positiue definite

~rrutrices. Then

Proof. Xote thnt both il',., and -qD are outer inverses of -4. It folIoss by Eq.(5. l) that

as required for Part (a). [7

Theorem 16.8. Let -4 E LwLXnL be giuen with Ind(,-l) = k. and M . E C''LX"' bt, two positive dcfirrite

( a ) r ( - 4 ; ~ : ~ + , ~ - -L..L~ ) = r - r ( - ~ ~ ) -

Proof. Note tha t both .4-4LfSN and dAD are idempotent. IL fohx r s from Eq.(S.l) that

as required for Part (a). Similarly we can show Part (b). Cornbining TIieorcm 16.G(a) and Thcoretil

LG.i'(a) ÿields Part ( c ) . 0

Theorem 16.9. Let -4 E Cm"" be giuen? M. Ar E CnLX"' be two positire dcfinite ntatrice.~. T f ~ e n

Proof. Follows from Eq.(4.1). (7

Based on the result in Theorem 16.3(b), we can extend the concept of power-EP mat rk to weighted

case: -4 square matris -4 is said to be weighted power-EP if both R(-49 ) R(3- ' -4 ' ) and R[(-4q-I

R(lLf.4) hold, where both 1.i and 1V are positive definite matrices.

Theorem 16.10. Let -4 E CmX" be given with Ind(A) = k, and 151, ,V E CnLXm be two positive definite

Proof. Follows from Eq.(4.1). CI

Theorem 16.11. Let -4 € Crnxn be given, M. S E und N . T f CnXn be /our positive definite

Proof. Sote that both A Y , , ~ and are outer inverses of -4. Thus it follows by Eq.(S.l) tliat

cistablishing Part (a). O

Theorem 16.12. Let -4 E CnLX" be given. l'LI. S f CnLX"'. Ar: T E Cn"" be four positive (lefinite

t Proof. Follows from Eq.(3.1) by noticing that .-LA!,,,,., AM,~.-I, . ~ r l L , ~ and ilsnT-4 are idernpotenr

niat rices. C!

Theorem 16.13. Let A E CrrLXm be un idempotent o r tripotent ~na tr ix . und J I . :\- E CniA"' hc t u;n

positive definite matrices. Then

(b) -4 = R(M.4) = R(NA) = R(.Aa), i.e.. -4 is weighted EP.

Proof. Note that -4, A:,, E -4{2) when -4 is idempotent or tripotent. It follows by Eq.(5.l) that

as reqiiireci for Part (a). O

Theorem 16.14. Let -4, B E CrnXrn be given, M , N E Cn'X"' be two positive definite rnutnces. T k n

Proof. Follows from Eq.(4.1).

Theorem 16.15. Let A f C n Z X m be yiven with Ind(i4) = 1. P. Q f C"'Xm be two no~~.sing7~lur n r at7-ices.

Proof. It is easy to verify that both ( P A Q ) ~ and Q-'-4#PV' are outer inverses of P.4Q. Tliixs ir follon-s

by Eq.(5.l) that

cstahlishing Part (a) and then Part (a). [3

Theorem 16.16. Let A E C m x m be yiven, Ad, 1V E Cmxnl be two positivc definite matrices. Thcn

Proof. It is easy to veriFy that both ( ~ - 4 Q ) t and Q - L ; L ~ . ~ P - ~ are outer inverses of P.4(2. Thus it

follows by Ecl.(5.l) that

cstablishing Part (a). 0

Chapter 17

Reverse order laws for weighted Moore-Penrose inverses of products of mat rices

Jus t as for Moore-Penrose inverses and Drazin inverses of products of matriccs. \Lee cari alsu corlsitler

reverse order laws for weighted Moore-Penrose inverses of products of matriccs. Noticing the basic k t in

Ecl-(lG.'L). we cari easiIy extend the resuIts in Chapter 8 to weighted lloore-Penrosct irivcrses of prortticcs

of niatrices.

Proof- The equivdence of Part (a) and Part (b) follows directly korn applying Eq.(16.2) to the both

sides of ( . - I B C ) ; ~ , ~ = c & . ~ ~ B ~ . ~ . ~ ~ ~ , ~ and sirnplifying. Observe thar tlie left-liancl side of Part ( b ) c-;iii

:ilso be mritten

( J [ ~ - ~ B C N - $ ) ' = [ ( ~ ~ t - ~ p - + ) ( p + ~ y - Q ) ( q f cLv-4)]t.

111 thüt case, tve see by Lemma 1.1 that Part (b) holds if and only if

Simplifying this rank equality by the given condition t h LU, !Y. P and Q are positive definite. wt3 oI,tairi

the r a d eqiiality in Part (c). Ci

Corollary 17.2. Let -4 f CmXn, B E Cnxk. and C E Ckxf he given and let .J = .ABC. Let P E Px"

t ~ n d Q E C k X k be two positive definite matrices. Then tire following three statements are equiualent:

(a) (ABC)' = c & , ~ B ~ . ~ I ~ : , ~ .

(b) (ABC)' = (Q+ c ) ~ ( P C B Q - ~ ) ~ ( A P - + ) t e

BQ-' B* PB O

(4 T O - J J S J JC-QC = r ( B ) i- r(.J).

-4 B AP-'-4-.J BC O 1 Proof. Follows frorn Theorem 2.1 by setting Af and LV as identity matrices.

Corollary 17.3. Let -4 E Cmxn, B E C n x k , und C E C k x L be given am Let

E CnLXmI N E C l X 1 be two positive definate matrices. Then the following three statements are eqxiurrler~t:

Proof. Follows frorn Theorem 17.1 by setting P ancl Q as identity matrices. 0

Corollary 17.4. Let .-I E Cm ", B E CnX\ and C E c k X 1 be giuen luith r(-4) = T L und r(C) = k. Let

d f E Ç r n x m . iV E CL"', P E Cnxn, und Q f CkX" be four positive definite matrices. Then the following

Proof. The given condition r(,-l) = n and r ( C ) = k is equivalent to -4t.4 = I r , . CCt = Ik. arid

r.(,-LBC) = r ( B ) . In that case. we can show by block elementary oprat ions tha t

are c!qiiivalcnt. the dctailed is omitted here. This result irnplies that

Thiis iinder the given condition of this corollary, Part (c) of Theorein 17.1 reduccs to

which is obviously equivalent t o Part ( c ) of this corolkuq.. O

CoroIIary 17.5. Let -4 E C m x m , B E Cmxn,and C E Cnx" be giuen with -4 and C rionsinyular. Let

il[, P E C m X n z and IV, Q E CnXn be four positive definite Hermitian matrices. Then

(a) D ABC)^^,,, = B L , ~ - A - ' a R( P-'.4'M.4B ) = R(B) and R[ (BCN- 'C=Q) ] = R ( B S ) .

(b) ( ; IBc )M, , = C-L Bf.4-L R(A*iZ/I-AB) = R ( B ) and R[(BCN-'Cm)' 1 = R(Bœ).

( c ) (.ABC)t = CL B L , ~ A - ' t) R(P-L-4*,IB ) = R ( B ) and R[(BCCaQ)' ] = R(BS) .

I r t partz'cular, the following two identities hold

Proof. Let -4 and C be nonsingular matrices in Corollary 1'1.4. We can obtain Part (a) of this corollary.

Parts (a) and ( b ) are special cases of Part (a). The equality (17.1) follows from Part (a} b. setting

P = -4'~Lf-4 and Q = (CN- 'C*)- ' . 13

Theorern 17.6. Let A E C n t X n , B E C n X k , and C E ckX' be giuen and denote J = -4BC. Let

JI E C r J ' X m I LV E C l X L , P E Cnx " , and Q E C k X k be four positive deJinite matrices. Then the following

two statements are equivalent:

(a) ( - 4 ~ ~ ) ! & . ~ v = ( B C ) L , N B ( - ~ B ) !ir,p -

[ B* (.-iB)*i\f .J B* PBC

(b) 1- .J!V-~(BC)* O = 7 - ( ~ ) + I ' ( . J ) .

ABQ-'B- O O

17.1. w e know ttiat

1 Proof. Mite .ABC as .ABC = ( . A B ) B L , ~ ( B c ) and notice tliat ( B ; . ~ ) Q , - , = B. Then Li- Tlicorcrii

IiolcIs if and onIy if

Xote by Eq.(l.5) that

Thus by block elenientarÿ operations, we can deduce that Eq.(17.3) is eqiiivalerit to Part ( c ) of t,he

theorem. The details are omitted. 13

Corollary 17.7. Let -4 E CmX". B E C n x k , und C E C k x L 6, giuen and denote ./ = .ABC. Let

E c ~ ~ ~ ~ ~ ~ . Ar E C L X L , P E Cnx and Q E C k x 6e four positive definite matrices. If

then the weighted Moore-Penrose inverse of the prodmt -4BC satzsfies the following two equalities

Proof. Under Eq.(l7.4), we know chat

which is equivdent to

R(BC) = R(B), und R(BaA4*) = R ( B œ ) -

Baed on them we further obtain

Cncler these two conditions, the left-hand side of Part (b) in Theorem 17.G reduces to 2r(B). Thus

Part (b) in Theorem 17.6 is indentity under Eq.(17.4). Therefore ive have Eq.(lÏ.5) under Eq.(l7.-1).

Consequently nriting -4BC as .ABC = ( -LB)B; ,~ (BC) and applying Eq.(lT.5) to it yields Eq.(lT.G).

0

Sonie applications of Corollary 17.7 are given below.

Corollary 17.8. Let -4, B € C m X n be given, fi1 E C"'%"', AT E C n X " , P E C'mX'"'~ und Q E C"""'" be

four positiue definite matrices. If -4 and B satisfij the rank additivity condition

then the ,wezghted Moore-Penrose of -4 + B satisfies the two eq?iulities

Then the condition Eq.(17.7) is equivdent to r ( U D V ) = r ( D ) . Thus it turns out that

which is exactlÿ Eq.(lï,S). Nest write -4 + B as

Tlien the condition Ec1.(17.7) is also equitdent to r(CJl D L k;) = r ( D I ) . Thus ic folIows by Eci-(I7.5) tli;lt

( L ~ I DI &)if-, = ( D ~ v~)&, ,D~ ( u ~ D~): ,~ , , , which is exactly Eq.(lT.9). a

-1 generaiization of Corollary 17.8 is presented below, the proof is ornitted.

CoroiIary 17.9. Let -Ai , - - - . -qk € Cmxn be giuen. und let E Cm "', 1V Cnxn , p E ~ k r r i x k n i . and

Q f CknX" bbe four positive definite Hermitian matrices. If

then the wezghterl Moore-Penrose inverse of the suni satisfies the following two epal i t ies

1 = 1 h l , IV

Corollary 17.10. Let A f C m X n . B E ~ r n x k , c E C I X ~ . ;1 E c l x k /je giuen. :\l, p E ~ ( ~ L + [ ) ~ I ~ ~ - ~ I

N. CS E c("+ ") "1 be four positive definite matrices. If

Proof. Under Eq.(17.13). we see that

TLius by Corollary 17.6. we obtain

d i i ch is esactly Eq.(17.14). When P = I,,,[ and Q = I,,+k, we have

- iv-& ' i=L "1"-il'] [;] - - [ [ O . O l t

Siniilarly we cari tleciuce

Piitting botii of tliern in Ecl.(lï.l4) yields Eq.(l ï . l5) .

References

[l] T. W- Anderson and G. P. H. Styan. Cochran's theorem, rank additivity and tripotent rnatriccs. In Statistics

and Probability: Essays in Honor of C. R. Rao (G. KalIianpur et al. eds.). North-Hoilrtnc!. Arusterdarri. 1982.

pp. 1-23.

[2] E- Xrghiriade. Remarques sur l'inverse généralisée d'un produit de matrices, Atti -4ccad. Saz. Lincei Rend-

Cl. Sci. Fis. Mat. Natur. 42(1967), 621-625.

[3] J. K. Baksdary <and T. Mathew, Rank invariance criterion anci its application to the iinified theory of l e a s

sqi~ares. Liriear Algebra Appl. 127(1990), 393-401.

[-LI .J. K. Bc-il;sdar~. F. Pukelsheim and G. P. H. Styan. Sorne propcrties of matrix p'utial orcicririp. Linear

AIgebra .Appl. 119(1989), 57-85.

[5j J . 2. Baksalary aud G. P. Styan. - r o u n d a formula for the rank of a matri-u prociuct with some staristical

;ippLicationç. Graphs. matrices. and deçigns, Lecture Notes in Pure and Appl. Math.. 139. Dckker. S c w kork.

1993. pp. 1-18.

[G] C . S. Bdlantinc. Products of EP matrices, Linear Algebra and Appl. 12(1975), 254-267.

[7] D. T. Bmvick and J. D. Gilbert, Generalization of the reverse order Iaw Mth related resultst Linear XIgebra

=\ppl. S(1974), 340-349.

[SI D. T. Barntick m d -1. D- Gilbert. On gencralizations of thc reverse orcier Law with rclated results. SI.U[. .J.

.-lppl. Math. 27(1974), 326-330.

[9] C. L. Bell, Generalizcd inverses of circulant 'and generalized circulant matrices. Liricar Algcbra Appl. 39( 1981 ).

133-142.

[IO] A. Ben-Israel and T. N. E. Greville. Generalized inverses: theory and applications. Corrccted reprint of tlic

197-4 origind. Robert E. Krieger Publisbing Co., Inc., Huntington, New York, 1980.

[l 11 J. BCrubé, R. E. Hartwig and G. P. H. Stym, On canonicd correlations and the dcgrees of non-orttiogoriality

in the thme-way layout. In Statistical Sciences and Data Analysis: Proceeciings of the Third Pacific Arta

Statistical Confcrcnce. Tokyo. L991 (K. Matusita ct al., cds.), \'SP Internatiorial Science Publislicrs. L t rech .

The Nctherlmds, 1993, pp. 247-232.

[13] S. L. Campbcll and C. D. Meyer, Jr. EP operators and generalized invcrçes, Cmad. h h t h . Bull. lS(1975).

327-333.

1131 S. L. Campbell and C. D. Meyer, J r , Generalized inverses of linear transformations. Corrcctecl reprint of the

1979 original. Dover Publications, Inc., New York. 1991.

[L4] D. Carlson, Liqiat are Schur complements, an-maya! Linear Algebra Appl. 74(1986), 234-275.

[15] D. Carlsori, E. Haynsworth and T. Markham, A generdization of the Scl~ur complement by rncans of thc

5foorcPerirose inverse. SIAM J. Appl. Math. 26( l974), 169-175.

[lG] J . Chen, A note on gencralized inverses of a product, Xortheast Math. .i. 12(1996), 431-440.

[Li] S. Chen and R. E. H,artwig, The group inverse of a triangular matrix. Lincar Algcbra --Ippl. 237/238(1996).

97-LOS.

[lS] Y. Chen? The generaiized Bott-Duffin inverse and its applications, Linear Algebra Appl. 134(1990). 71-91.

[19] Y. Chen and B. Zhou. On the g-inverses and nonsingulrtrity of a bordered matrix [ ] . Liuear l g e b r a

Xpp1.133(1990), 133-151.

['LOJ R. E. CIiue, Note on the generalized inverse of the product of matrices. SIAM Rev. G(1964), 57-58.

[21] R. E. Clinc. Representations of the generalized inverse of silrns of matrices. SI.-\M .J. Xiirner. Anal. Scr B

2(1965). 99-114-

[22] R. E. Clinc, Inverse of rank invariant powers of a matrix. SIAM J . Numer. Anal. 3(1365). L52-197.

[23] R- E. Cline and R- E- Funderlic, A tkeorem on the rank of a difference of matrices. 1.~11. Amer* 5lath. Soc.

52(1976). 45.

[24] R. E. C h e arid R. E- Funderlic, The rank of a difference of matrices and associated generalizcd inverses.

Linear Algebra Appl. 24(1979), 185-215.

[25] R- E. Cline and S. N. E. Greville, Xa extension of the generalized inverse of a niatriu. SIAS1 J . Xppl. 5Iath.

L9( 1972), 682-688.

(261 P. .J. Davis, Circulant Matrices, Wiley. New York, 1979.

[27] A. R. De Pierro and M. Wei, Reverse order law for refiexive generalized inverses of products of matrices.

Linear Algebra Appl. 274(1998), 299-311.

[-SI 1. Erdelyi, On the 'reverse order Iaw7 related to the generdized inverse of matris products. J . Assoc. Compiit.

.\.lach. 13(1966), 439-433.

[29] J . A. Fil1 and D. E. Fishkind, The 3Ioore-Penrose generalized inverse for srims of matrices. SLAM .J. AppI.

Irath.. to appear.

[30] A. .J. Getson anc l F. C. Hsuan, (2)-Inverses of Matrices and Thtbir Statistical Applicatioris, Lecture Sotcs i r i

Statistics 47, Springer, Berlin, 1958.

f31] 41. C. Gouvrcia and R. Puystiens. About the group inverse and Moore-Penrosc irivcrsc of a product. Linear

XIgebra Xppl. 150(1991), pp. 361-369.

[32] T. N. E. Greviile. Xote on the generalized inverse of a matris product. SI-441 Rev. S(1966), 525-531.

[33] T. S. E. Greville. Solutions of the matrix equation S.4-Y = S. SL4M J. Appl. Math. 26(1974). 825-832.

[34] .J. Gross. Some remarks concerning the reverse order Iaw, Discuss. Slath. -4lgelirn Stocliast ic kIetliods

17(1997), 135-141-

[351 .J. Gross cuitl G. Trcnkler, Gencralized and hypergeneralized projcctors. Lincar Algc1)ra App1. '161(1997).

463-474.

[36] -1. Gross and G. Trenkler, Xonsingularity of the Difference of two oblique projectors, SIXM J . \ latris Anai.

Xppl. to appear.

[37] R. E. Hartwig, The resultant and the matris equation -4-Y = S B . SLIM .J. Xppl. lfath. 22(1972). 538-54-1.

[3S] R. E. Warcwig, Block gencralized inverses, Arch. Rational Slech. .inal. Gl(1976). 197-251.

[39] R.. E. Hartwig, Singular value decompositions and the MoorcPenrose inverse of bordered matrices. SIXM J .

XppI. Math. 31(1976), 31-41.

[40] R. E. Hartwig, Rank factorization and the Moore-Penrose inversion. J. Iridust. Matli. Soc. 26(197G), 49-63.

R. E. Hartwig, The reverse order law revîsited. Linear Algebra Appl. 76(1986). 241-216.

R. E. Hartwig and 1. 3- Katz, On products of EP matrices, Lincar Xigebra Xppl. 252(1997), 338-345.

R. E. Hartwig, M. OrnladiC, P. Semrl and G . P. H. Styan, On sGme characterizations of painvise s t ~ v

orthogonality nsing rank and dagger additivity and subtractivity. Special issue honoring C. R. Rao. Linear

Xlgebra -4ppl. 237/238(1996), 499-507.

R. E. Hartwig and K. Spindelbikk, Partial isometries, contractions and EP matrices, Linear a d .IliiltiIinca.r

Algebra 13(1983), 295-310.

R. E. Kartmig and K. Spincielbiick, Matrices for which .A' and can cornmute. Lincar and Multiliriear

Xlgebra L4(1984). 241-256.

R. E. Hartwig and G. P. H. Styan. On some characterizations of the "star7' partial orticring for matrices and

raik subtractivity, Linear -1lgebra Xppl. 82(1986), 115-161.

R. E. Hartwîg and G. P. H. Styan, Partidly ordered idempotcnt matrices. In Proceedings of the S e c u ~ i d

International Tampere Conference in Statistics (T. Pukkila and S. Puntanen. ccis.). 1987. pp- 361-383.

E. V. Haynsworth, Applications of an inequdity for the Sdiur complement, Proc. Amer. Math. Soc. 24(1970),

512-516.

D. -4. Harville. Generdized inverses and ranks of modified matrices. .J. lndian Soc. Xgricultiiral Statist.

49(1996/1997). 67-78.

R. A. Horn and C. R. Johrison, Topics in matriv analysis, Cambridge L~nivcrsity Press. Cambridge. 1991.

F. C. Hsuan, P. Langenberg and -1. J. Getson, The (2) inverse with applications to statiscics. Liricar Algebra

-4 c C. H. Hurrg and T. L. Slarkham, The IvIoore-Penrose inverse of a partitioned matris 1 1 . ~iiiciir

Li -I

Algcbra Appl. 1 l(lgT5), 73-86.

C. H. Hung ancl T. L. blarkham, The Moore-Penrose inverse of a sum of matrices, .J. Austral. l iath. Soc.

Scr. A 24(1977). 385-392.

E. P. Liski and S. Wang, On the (2)-inverse mtl sorne ordering propertics of norinegative dcfiriite rriatrices.

Acta hiatli. Xppl. Sinica( English Scr.) i2(1996). 8-13.

G. hlzusaglia and C. P. H. Styan, 1Vlien does rank(.-l + B) = r;rrik(-4) + rcuik(B)'! Canad. hIatli. Bull.

l5(l!lï2), 451-452.

G. hiarsaglia and G. P- H. Styan, Equaiities and inequalities for ranks of matrices. Lincar and ~Itiltilincar

Xlgebra 2(1974), 269-292.

G- SIarsaglia arid G. P. H. Styan, Rank conditions for generdized inverses of partitioneci matrices. SiiilkhyZ

Scr. A 36 (LSTLL), 437-442.

A. R. M e e n k h i and R. Indira, On sums of conjugate EP matrices. Irrdian .J. Piire Appl. Slatii. 23( 1932).

179-L84.

A. R. Meenakshi and R. Indira, Conjugate EP, factorization of a matris. Math. Stiidcnt 61 (199'2). 136 - 144.

A. R. Meenakshi and R. Indira, On products of conjugate EP, matrices, Iiyungpook Math. J . 32 (1992).

103-1 10.

[ ~ l ] A. R. Meenakshi and R. Indira, On Schur complements in a conjugate EP rnatrix, Studia Sci. Math- Hungar.

32 (1996),31-39.

[62] .A. R. Meenakçhi and R. Indira, On conjugate EP matrices, Kyungpook Math. J . 37(1997). 67-72.

[63] -4- R. hleenakshi and S. Krishnamoorthy, On k-EP matrices, Linear Algebra Xppl. 269(1998), 219-232.

[G-L] A. R. hiecnakshi and C. Rajian. On sums and products of star-dagger matrices. J. Inclian Math. Soc. 50(19SS).

l+l- 156.

[65] C. D. Meyer, J r , Generalized inverses of block triangular matrices, SEXM J . Math. Appl. 19(1970). 741-750.

[66] C. D. Meyer, Jr , Some remarlcs on EP, matrices. and generalized inverses. Linear Xlgebra and Xppl. 3 (1970).

275-278.

[67] C. D. Meyer, .Jr, The Moore-Penrose inverse of a bordered matrix, Linear XIgebra Appl. 5(1972). 375-382.

[G8] C. D. Meyert Jr, GeneraIized inverses and ranks of block matrices, SIAM J. -4ppl. Math. 23(1973), 597-602.

[69] .J. Mao, The Moore-Penrose inverse of a rank modified matrix. Xiimer. Math. .J. Cliinese L7niv. 11( 1989).

35: -36 1.

[XI] J . Miao, General espression for the Moore-Penrose inverse of a 2 x 2 block matris. Lincar Algclira Appl.

151(1990), 1-15.

[T l ] S. K. Mitra, Tlie matrix equations -4-Y = Cl -YB = D, Linear Algebra -4ppl. 59(1954). 171-Ml.

[Ï2] S. K. Mitra. Properties of the fundamental bordered matriv used in Iinear estimation. In Statistics a i c i

Probability, Essays in Honor of C. R. Rao (G. KaIlianpur et al, e h . ) , North Hollarid. Kew York. 1982.

1731 S. K- Mitra, Noncore square matrices miscellany. Linear Algebra Appl. 249(1996). 47-66.

[74] S. K. Mitra and R. E. Hartwig, Partial orderings based on the oiiter inverses. Linear Xlgcbra Appl. 1 7G( 1992 ).

3-20.

[XI Sv K. Mitra and P. L. Odell, On pardiel summability of matrices. Liuear Xlgcbra Appl. 14(19S6). 239-255.

[7G] Al. Z. -T,?çhetl and S. Chen. Convergence of Newton-like rnethods for singiilar operator cqiiations iisirig oritcr

inverses, Ntrmer. Math- 66(1993). 235-237.

[77] FV. V. Parker, The mat ri^ equation -4-Y = -YB, Duke Math. J. 17(1950), 43-51.

[7S1 M. Pc,vl, On normal and EP, matrices, Michigan Math. J. 6(1959), 1-5.

[79] hi- Pearl. On normal EPr matrices, Michigan Math. J. S(1961). 33-37.

[SOI R. Penrose, .A generaiized inverse for matrices, Proc. Cambridge Philos. Soc. 5 l(lXi5). -LIH--Il3.

[SI] S. Piintanen, P. s e m l and G. P. H. Styan, Some rernarks on the parallel siim of two matrices. In Procccdings

of the A. C. Aitken Centenarÿ Conference, Dunedin, 1993, pp.243-256.

[52] C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wilcy, New York. 1971.

[83] C. R. Rao and H- Yaiiai, Generalizcd inverses of partitioned matrices useful in statistical applications. Lincar

AIgelra -4ppl. ÏO(lgS5), 103-1 13.

[S4] Robinson, Donald W. Nullities of siibmatrices of the Moore-Penrose inverse. Linear Algcbra Xppl. 94(19ST).

127-132.

[85] S. R. Searle, On inverting circulant matrices, Linear Algebra Appl. 25(1979), 77-59.

[S6] N. Shinozaki and M. Sibuya, The reverse order law (AB) - = B--A-, Linear Algebra Appl. O(1974). 29-40.

[Sï] S. Shinozaki and M. Sibuya, Further results o n the reverse order law, Linear Algcbra Appl. 27[1979). 9-16.

[Sa] S. Slavova, G. Borisova and 2. Zelev, On the operator equation .-LX = -YB. Cornplex analysis and applications

'87 (Varna, 1987), 474476, Bulgar. Acad. Sci.. Sofia, 1989.

[S9] G. P. W. Styan and A. Takemura, Rank additivity and matriv pol-vnomials. Studies in econometrics. tirne

series, and multivariate statistics, 545-558, Academic Press, New York-Loiidon, 1983. pp. 545-558-

[90] IV- Sun cznd Y. Wei, Inverse order rule for nreighted generalized inverse. SL4M J. Matris .Anal. Xppl. 19( 1998).

72-775.

[91] Y . Tian, The Uoore-Penrose inverse of a partitioned rnatrix and its applications. Xppl. Chincse 1Iath. . J .

Chinese University 7(1992), 310-314.

[92] Y. Tian, The Moore-Penrose inverse of a triple matrk product. Math. in Theory nnd Practice l(1992). Gd-10.

1931 Y. Tian. Calculating formulas for the r a d s of submntrices in the AIoorePenrose iriverse of a niatrix. J .

Beijing Polytechnical University 15(1992), 84-92.

[94] Y. Tian, Reverse order laws for the generalized inverses of muitiple matriu products. Linear Algebra Appl.

21 l(l994), 185-200.

[95] Y. Tian, The Moore-Penrose inverses of m x n block matrices and tlicir applications. Lineu ,Algebra Appl.

253(1998), 35-60.

[96] Y. Tian, Universal similarity factorization equalities over real Clifford algebras. Atlv. Appl. Clifford algebras

S( LXIS), 365-402.

[Of] Y. Tian. Universal similarity factorization eqrialitieci for rcal quaternion matrices. subrnitced for prihlication.

f9S] Y. Tian and G. P. H Styan, Rank of matrix expressions involving generalizcd irivcrses of matrices. Papcr

prescnted a t The Seventh International \Vorkshop on Matrices and Statistics, Fort Lauderdale. Florida.

DeceInber 11-14, 1998.

[99] M. Wei, Equivdent conditions for generdized inverses of products, Lincar AIgebra Appl. X(199T). 347-363.

[IO01 Y. Wei, A ciiaraccerization and representatian of the gencrdized ixivcrse -45::. and its application, Liricar

-4lgebra Xppl- 280(1998), 87-56.

(1011 Y. Wei and G. Wang, A survey on the generalized inverse =L:~T. In Proccedings of Meeting on AIatris

Xnalysis and Xpplications. Spain, 1997, pp.421-425.

[102] H. Werner, When is B-=i- a gcneralized inverse of -AB? Liriear Xlgebra Appl. 2FO(L994), 255-263.

(1031 H. Werner, G-inverses of m a t r k products. In Data Analysis and Statistics Inference (S. Schach and G.

Trenkler, eck.), Eul-Verlag, Bergisch-GIadbach, 1992, pp. 531-546.

Il041 E. A. Wibker, R. B. Howe. and J. D. Gilbert. E-qlicit solution to the reverse ordcr Iaw (-AB)+ = B,;,-4,.

Liriear Xlgebra Xppl. 25(1979), 107-114.