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Numer. Math. 14, t4--23 (t969)
Condition Numbers and Equilibration of MatricesA. VAN DER SLU
IS
Received March 6, 1969Introduct ion
In nu me rica l l inea r alg ebra one m eet s co nd it ion nu mb
ers [[AII []A-11[and s imi la rqu an tit ies su ch as (m ax
[a,i[)[[A-1H an d [[Ai[[ [[A-l[], wh ere A = (a,i) an d A i is the/
' - th co lumn of A . The norms a re ve ry d ive rse .
The problem then i s to de te rmine a row- and/or co lumn-sca l
ing of A whichm i n im i z es t he qua n t i t y unde r c ons ide r
a ti on .
I t i s the purpose of th is p aper to spec i fy a c lass o f
such qu ant i t i e s fo r whichthose scal ings can be given expl
ic i t ly . The resul ts wi l l be extens ions of someresul ts in
[2] . They wi l l a lso hold for non-square matr ices . Al l proofs
wi l l bec om pl e t e l y e l e m e n t a r y .
Also, in some cases where the min imiz ing scal ing can not be g
iven expl i c i tly ,i t can be sa id how fa r a t mos t fo r a ce
r t a in sca l ing the qua nt i t y un der cons ide ra t ionm a y b
e a w a y f r o m it s m i n im u m .
Convent ions~c2~ wil l deno te the se t o f rea l o r com plex m
n mat r i ces , m>=n, and A wil l
a l w a ys be a n e l e m e n t o f ~ , ~ . A n wil l denote the
t ransposed of .~. ~,n and ~wi ll denote th e c las s o f non-s
ingula r rea l o r complex m m or n n d iagona lmat r i ces .
X and Y wi l l a lways denote rea l o r complex ca r t es i an
spaces of d imens ion nan d m res pe cti ve ly, an d wi th no rm s
][. []a an d ][. I[~ res pe cti ve ly.
Al l of ~X,~n, ~,n, ~ , X an d Y wil l be real or a l l of th em
will be complex.Thi s induces the quant i t i e s
lub~a(A)=maxtlAx[[Jl[x[~ a n d g l b ~ a ( A ) =min [A ~ILIII-II~
fo r a n y A E~J~m~.x*0 lub m and glb m, p and q real num ber s or
o% will den ote the case tha t [[. It~an d [[. [~ are th e H61d er
p - an d q-no rm resp ectiv ely, def ined as [[xIlp= (2 [xj[p) l
/p,and s imi la rly for q.
[]A [[p = lu bpp (A ).[x[ wil l deno te the vec tor whose coord
ina tes a re the modul i o f the cor responding
coord ina tes o f x ; s imi la rly for a ma t r ix [A I.A ve c t
o r xEX, x4=O, will be called a maximizing vec to r for A wi th
respec t
to the given norms i f [ [A xl[,,/[[x[~=tub~,a(A ). Also, x will
be called a minimizingvector if ][Ax[IJ[[x[[a= glb~,~ (A ).
Def. wiU indicate a def ini t ion, Th. wi l l indicate a smal l
or intermediate resul t .
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Condition Numbers and Equilibration of Matrices 172.
Equilibration Theorems
Def. 2.1. In the following P will be a ma tri x wit h m rows if
B is left-multipliedb y DE~,~, B wilt have n columns if B is rig ht
-m ult ipl ied b y DE~)~. A lso, If. I~wi ll denote an y vec
tornorm.
Def. 2 .2 . For a ny two matr ices A an d B and an y two m atr
ix func t ions ~band ~ we def ine ~(B, A) =~v(B)/~(A).i f the r
ight hand side exists .
We then h ave the fo llowing two theoremsTheorem 2.3. Column
Equil ibrat ion Theorem. I f w(B)=max I[Bi[b (where B itdeno tes
the ] - th colum n of B) and ~ is r ight-mono tonic on A ~D, and
D6~D, is such
th at B/~ is colum n-equil ibrated in the sense of i t ( ie all
columns of B/5 haveequal y-norm). Then(a) ~ (B /5 , A D) ---- ra in
~ (BD, A D)(b) An y matr ix D for which the m inimu m in (a) is a t
ta ined m ay be obta ined
by m ul t ip ly ing /~ by a d i a gona lm a t r ix whose d i a
gona l e l e m e n t s ha ve e qua lmod ulus if and onl y if $ is s
trong ly r ight-m onoto nic a t A/5 (hence in this casecolumn
-equil ibrat ion of th e arg um ent of ~p is a lso a necessary cond
it ion for them in im u m to be a t t a ine d ).
Proo/. (a) I t is no restr ic t ion to assume that D=I , i .e .
that B alreadyis column -equil ibrated. The n m ax [Ie oj l lv : m
ax ]di i I. m ax IIBjI/r and (A D) --
