An equivalent reduction of a 2-D symmetric polynomial matrix N. P. Karampetakis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki

An equivalent reduction of a 2-D symmetric polynomial matrix

N. P. KarampetakisDepartment of MathematicsAristotle University of ThessalonikiThessaloniki 54006, GreeceEmail : [email protected] URL : http://anadrasis.math.auth.gr

Contents

Preliminaries Problem statement for 1-D polynomial matrices

Finite and infinite elementary divisor structure of 1-D polynomial matrices Problem statement and solution

Problem statement for 2-D polynomial matrices Invariant polynomials & zeros of 2-D polynomial matrices Zero coprime equivalence transformation and its invariants Zero coprime system equivalence and its invariants Problem statement

2-D symmetric polynomial matrix reduction procedure 2-D polynomial system matrix reduction procedure Implementation in Mathematica Conclusions

Problem Statement – 1-D polynomial matrices

Definition [Vardulakis, 1991] Let ( ) [ ]p mT s R s , with ( ) ( ) min ,srank T s r p m .

Then there exist unimodular matrices ( ) [ ] , ( ) [ ]p p m mL RU s R s U s R s (i.e.

det[ ( )],det[ ( )] \ 0 ),L RU s U s such that

( ) 1 ,( ) : 1,1,..,1, , ,..., ,0CT s L R z z r p r m rS s U s T s U s blockdiag f s f s f s

with 1 z r and 1/i if s f s , 1,..., 1i z z r . ( ) ( )T sS s is called the Smith form

of ( )T s (in ), where if s R s are the invariant polynomials of T s . Assume

that the partial multiplicities of the zeros ,i i k are , , 1 ,0 i z i z i r i.e.

, ˆ ˆ( ) ( ), , 1,.., with ( ) 0i j

j i j jf s s f s j z z r f

The terms ,i j

is are called finite elementary divisors of ( )T s at is . We also

denote by n the sum of the degrees of the finite elementary divisors of ( )T s i.e.

,1

: degr k r

j i jj z i j z

n f s


Definition [Praagman] Define the dual polynomial matrix T s of T s as

10 1

1:

p mq q qqT s T s T s T s T R s

s

Let ( ) , ( )p p m m

L RU s R s U s R s be rational matrices having no poles and zeros at

0s and such that

1 20,( ) ( ) , ,..., ,0r

L R p r m rT sU s T s U s S s diag s s s

where 0T sS s is the local Smith form of T s at 0s . The terms js are called

the infinite elementary divisors of ( )T s i.e. are actually the finite elementary divisors of the dual matrix of ( )T s at 0s We also denote by the sum of the degrees of the infinite elementary divisors of ( )T s i.e.

1

:r

jj

Definition Similarly, we can find ( ) ( ) , ( ) ( )p p m mL RU s R s U s R s having no poles

and zeros at the point 0s such that

0( ) 0 0 ,( ) : 1,1,..,1, ,..., ,0z r

T s L R p r m rS s U s T s U s blockdiag s s

In that case, 0( ) ( )T sS s is called the local Smith form of ( )T s at the local point 0s .

Problem Statement – 1-D polynomial matricesProblem 1. Given the polynomial matrix

11 1 0

r rn nn nT s T s T s T s T s

find a matrix pencil nr nrsE A s

with the same finite and infinite elementary

divisor structure with T s .

Solution.

Step 1. Find a matrix pencil nr nrsE A s

.

Step 2. Show that there exists a “transformation” between T s and sE A that

preserves the finite and infinite elementary divisor structure.

Problem Statement – 1-D polynomial matricesStep 1. First companion form

1 2 0

0

0

n n n

p p

p p

sT T T T

I sIsE A

I sI

- -+é ùê úê ú-ê ú- = ê úê úê úê ú-ë û

L

O M

M O O

L

Step 2. Show that there exist a) either unimodular transformations (or extended unimodular equivalence transformations) of the form:

1

0

0 n r

T sU s sE A V s

I

preserves the f.e.d. structure

and

1

0

0 n r

T sU s E sA V s

I

preserves the i.e.d. structure

where 11 1 0

n nn nT s T T s T s T s

is the dual matrix [Goghberg et. al. 1982]. b) or a strict equivalence transformation that connects T s and sE A [Vardulakis and

Antoniou 2003], c) or a divisor equivalence transformation that connects T s and sE A

[Karampetakis and Vologiannidis 2003].

Problem Statement – 1-D polynomial matricesProblem 2. Given the symmetric polynomial matrix

11 1 0 ,

r rn n Tn n i iT s T s T s T s T s T T

find a symmetric matrix pencil , ,nr nr T TsE A s E E A A

with the same finite

and infinite elementary divisor structure with T s .

Motivation Numerical methods that ignore the special structure of the

polynomial matrix T(s) (like the companion form) often destroy these qualitatively important spectral symmetries, sometimes even to the point of producing physically meaningless or uninterpretable results.

Storage and computational cost are reduced if a method that exploits symmetry is applied.

i.e. The solution of Ax=B, with A symmetric, via a symmetric banded solver uses O(n) storage and O(n) flops, while using LU methods that not exploits symmetry uses O(n2) storage and O(n3) flops.


Solution of Problem 2 Higham et.al. 2005 Symmetric linearizations for matrix polynomials.

The reduction is used for the solution of the polynomial eigenvalue problem T(s)x=0.

A vector space of symmetric pencils sE-A is generated with eigenvectors closely related to those of T(s).

No transformation is used.

The matrix pencil proposed by Antoniou and Vologiannidis is not in the vector space of symmetric pencils proposed by Higham et.al..

Antoniou and Vologiannidis 2004. Linearizations of polynomial matrices with symmetries and their applications.

One specific symmetric linearization is proposed. The new matrix pencil sE-A is connected with T(s) through a unimodular equivalence relation.

Problem Statement – 2-D polynomial matricesDefinition: Given a p q polynomial matrix ,P s z , the ith order invariant polynomial

,i s z of ,P s z is defined by :

1

,if 1

,, :

0 if min ,

i

ii

D s zi t

D s zs z

t i p q

where t is the normal rank of ,P s z , 0 , 1D s z and ,iD s z is the greatest common

divisor of all the i i minors of the given matrix ,P s z . ,iD s z is called the ith

determinantal divisor of ,P s z . The zeros of ,iD s z are called the ith determinantal

zeros of ,P s z .

The i-th order invariant zeros of ,P s z are the elements of the variety PiV I

defined by the ideal PiI generated by the i th order minors of ,P s z .

Definition: Let 1 2, ,..., sS f f f be a set of polynomials in ,F s z .

1 2 1, ,..., , ,

s

s iiI f f f F s z f F s z

is called the ideal generated by 1 2, ,..., ,sf f f F s z , or by S . The variety of I, defined

by V I is the set of all solutions in 2F of the system of algebraic equations

0, 1,2,..,if i s , and will be denoted by

2 , 0, 1,2,...,iV I a F f a i s

Problem Statement – 2-D polynomial matricesExample: Consider the 3-D polynomial matrix

2

2

0, ,

0

x xyP x y z

z xz

The invariant polynomials are the following

10 1 1

0

21 2 2

1

,, 1, , 1 , 1

,

,, 1, , ,

,

D s zD s z D s z s z

D s z

D s zD s z D s z xz s z xz

D s z

The ideals generated by the first and second order minors are

3 2 2 22

2 21

2 1

, ,

, , ,

P

P

P P

I x z x z xyz

I x xy xz z

I I

2 1 1 2 2 2 1 2 1

1

2 1

0, , , , ,0 , , , ,

0, ,0 ,

P

P

P P

V I y z x y x y y z

V I y y

V I V I


Problem 3. Given the symmetric polynomial matrix

1, , 1 0,0 , ,, , ,

r rp q p q Tp q p q i j i jT s z T s z T s z T s z T T

find a symmetric matrix pencil 1,1 1,0 0,1 0,0 , ,, ,pqr pqr T

i j i jszA sA zA A s z A A with

the same determinantal and invariant zeros with ,T s z .

Zero Coprime Equivalence transformationLet ,m lP , be the set of r m r l 2-D polynomial matrices where

max ,r m l .

Definition: [Pugh et.al. 1998] If there exist polynomial matrices , , ,L s z R s z such

that 1 2, , , ,L s z P s z P s z R s z

where the compound matrices

2, ,L s z P s z and

1 ,

,

P s z

R s z

have full rank 2, Cs z then 1 2, , , ,P s z P s z m lP are said to be zero-coprime

equivalent (ZC-E).

Invariants of ZCELemma: [Pugh et. al. 1996] Suppose that two polynomial matrices 1 ,P s z and

2 , ,P s z m lP , are related by zero coprime equivalence and let 1 1 1

1 2, ,...,P P Ph ,

where 1 1min ,P Ph r m r l and RPr rank P , denote the invariant

polynomials of 1 ,P s z and 2 2 2

1 2, ,...,P P Pk , where 2 2min ,P Pk r m r l ,

denote the invariant polynomials of 2 ,P s z , then 1 2P Ph i i k ic

for 0,1,...,max 1, 1i k h , where 1 21, 1P Pj j

for any 1, R \ 0 .ij c

Lemma: [Pugh et.al. 2005] Suppose that two polynomial matrices 1 ,P s z and

2 , ,P s z m lP , are related by zero coprime equivalence and let 1PjI for

1 11,..., min ,P Pj h r m r l denote the ideal generated by the j j minors of

1 ,P s z and 1PiI for 2 21,..., min ,P Pi k r m r l denote the ideal generated by

the i i minors of 2 , .P s z Then 1 2 , 0,1,...,P Ph i k iI I i h

where min 1, 1h k h and 1 21 or 1 in case or P Ph i k iI I i h i k

Zero Coprime System Equivalence transformation

Lemma: [Pugh et.al. 2005] The transformation ZC-SE preserves the transfer function of , , 1,2iP s z i , the determinantal, and invariant zeros of the matrices

,, , , , , , ,

,i

i i i ii

T s zT s z P s z T s z U s z

V s z

for i=1,2.

Definition: [Pugh et. al. 2005] Let two 2-D systems described by the Rosenbrock system matrices of the form

, ,, , 1,2

, ,i i

ii i

T s z U s zP s z i

V s z W s z

If there exist polynomial matrices , , , , , , ,L s z R s z X s z Y s z such that

1 2

1 1 2 2

1 1 2 2

0

0P P

T U T UL R Y

V W V WX I I

where the compound matrices 2, ,L s z T s z and 1 , ,TT T

T s z R s z have full

rank 2, Cs z then 1 2, , , ,P s z P s z m lP are said to be zero-coprime system

equivalent (ZC-SE).

2-D polynomial matrix reduction procedure

Consider the two variable polynomial matrix:

0

1, ,0 0, 0, 1 0,0, [ , ]

p

q p q q r rp q p q q

T z T z

T s z T z T s T z T z T s z

Without loss of generality we assume that p and q are always odd numbers (otherwise we add extra zero leading terms). Let

1

1

0 01

,

0 0

0 0 , , 1,2,..., 10

0 0

,

p p rp

r p k

k rk k k

rr k

r p

A z diag T z I

IT z I

A z C z C k pI

I

A z diag I T z

2-D polynomial matrix reduction procedure - ExampleExample: Consider the two-variable polynomial matrix

2 2, R ,r rT s z Ks Mz s z

where ,K M are symmetric real matrices. This polynomial matrix is associated with the 2-D discrete-time AR-representation

2 , , 2 0Ku x h y Mu x y k

Rewrite ,T s z as follows

2 01

3

3 2 2,, 0 0r r

T z T zT zT z

T s z s K s s Mz

We define the matrices

3 3

3 2 1

22 2 2

2 1

3 1 1

11 1 1

1 1

0 03 1

,

0 0 0

0 0 0 0 , 0

0 0 0 0

0 0 0 0

0 0 0 0 , 0

0 0 0 0

,

rp

r rr

rr

rr

r rr

rr

rr

r

A z diag T z I

I K IT z I

A z C z I CI

I I

I IT z I

A z C z I CI

I I

A z diag I T z dia

22 ,rg I Mz


Theorem: ( , ) [ , ]r rT s z s z and , [ , ]pr prsP s z s z are ZC-E.

Let

1 2 0

1 0

0,

0

p p p

r rs p p

r r

sT z T z T z T z

I sIP s z sA z A z A z

I sI

be the companion form of ,T s z in terms of s .

Proof: ( , )T s z and ,sP s z are connected with the ZC-E transformation

11 2 0

2

( 1) ,

( , )

, ( , )

( ) ( ) ( ) ( )

0( , )

0

0

s

qp p p r

qr r r r

q r r

M s z r r r

P s z N s z

sT z T z T z T z s II I sI s I

T s z

I sI I

, ,sT s z P s z


1

2

( , )

( , )

( , )

qr

q

r

r

T s z

s IT s z

s IN s z

I

are zero coprime

01 2

0( , ) ,

0

0 0

r p p p

r

r

r

r

s

T z

M s z P s

I sT z T z T z

I sI

sI

z

I

are zero coprime

2-D polynomial matrix reduction procedure - Example

2 01

3

3 2 2,, 0 0r r

T z T zT zT z

T s z s K s s Mz

( , ) [ , ]r rT s z s z and , [ , ]pr prsP s z s z are ZC-E.

The first companion form (by taking 3 0T z ) of the matrix ,T s z will be the

following

2

3 2 1 0 0

, 0 0

0 0s r r r r

r r r r

sT z T z T z T z K Mz

P s z I sI I sI

I sI I sI

which is not in symmetric form.

2 2

( , ) , ( , )

0

0 ( , ) 0

0 0

s

r r

r r r

r r r

M s z P s z N s z

I K Mz s I

T s z I sI sI

I sI I


Theorem: ,sR s z is is a self-adjoint polynomial matrix pencil which is ZCE to

,sP s z and therefore to ,T s z .

Let the following companion form of ,T s z

4 5

2 3

0 1

, :

0 0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0 0 0

0 0 0 0

s odd even

r

r r

r r

r r

r r

r

R s z sA z A z

sI

sI T z sT z I

I sI

sI T z sT z I

I sI

sI T z sT z

where

0 2 1

1 1 12 3 1

even p

odd p p

A z A z A z A z

A z A z A A z A z

, ,, s sT P s z Rs z s z

2-D polynomial matrix reduction procedure - example

Define the following companion form

13 1 0 2

2

0

, : 0

0

r

s r r

r

K I

R s z s A z A z A z A z I sI

sI Mz

The zero-coprime equivalence that connects the matrix pencils ,sR s z and ,sP s z

will be the following

1 1

2

2

, , ,,

0 0 0 0 0 0

0 0 00

0 00

0

0 0

0 0

ss

r

r r

M s z R s z N s zP s z

r

r r

r r

r

r r r

K

I sI

I sI I

K Mz I

I sI K

K Mz

I

I

I sI


( ) ( )

2 3 1

4 5 3 2 1 0

24 5 3 4 5

,,

0 0 0

0 0 0

0 0 0

0

0 0

0

0 0 0

0 0

0 0 0

0 0

0

0

0

0

0 0

0

sP s zM s z

I sI

I

T sT T T T T

T sT T sT s T

s

I

I

I T sT

I

I sI

I sT I

+é ùé ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úê ú+ + + ê úê úë ûë û

-

-

-

+ -144444444 2144444444444444444444424444444444444444444443

( )

( )

4 5

4 5

2 3

4 5

0 1

,,

0

0

0 0 0 0 0 0

0 0 0 000

0 0

0

0 0 0 00 0

0 0 0 00

0 0 0 0 00

s N sR s z

T sT I

T sTI I

IT s

I

sI IT

s T sTI sI

T sT IsI

r

=

+é ùé ùê úê úê úê ú+-ê úê úê úê ú+= ê úê úê úê ú

+-ê úê úê

-

úê úê úê ú+ ê úê úë ûë û

-

44444 44444444444443

144444444444444444424444444444444444443( )z

1444444444442444444444443

5 4 3 25 4 3 2 1 0,T s z T z s T z s T z s T z s T z s T z


( )

2 2 36 7 5 6

23 4 5 2 3 17 4 5 6 7

,

0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 00 0

0 0

0 0

M s z

T sT T

I

I

I

I T ssT T s T T sT Ts T T sT s T s T

é ùê úê úê úê úê úê úê úê úê úê úê ú

+ + +

ê úê úê ú+ + + + + +ë û14444444444444444444444444444444444444444424444444444444444444444444 3

( )

6 7 5 4 3 2 1 0

,

6 7

4 5

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0

0 00 0 0

0 0

0

0 0 0

0 0 0 0 0

0 0 0 0

0

0 0

0

0

0

0

sP s z

I sI

I sI

I sI

I sI

I sI

T sT T T T T T T

T sT

I sI

I

sI

I s

T IsT

I

+é ùê úê úê úê úê úê úê ú=ê úê úê úê úê úê úê úë û

+

-

+

=

-

-

-

-

-

-

-

-14444444444444444424444444444444444434444444444444444

( )

( )

( )

6 7

6 7 4 5

2 3

26 7

0 1

,

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 00 0

0 0 0 0 0 00 0 0

0 0 0 0 0

0 0

0 0 0

0

0 0

0

0

0 0

sR s z

sI I

sI

I

T sT

I

s T sT T sTI sI

IT sT

I sI s T sT s

T sT

é ùê úê ú +ê úê úê úê ú

+ +-ê úê úê ú+ê úê ú- +ê úê úê ú+ë û

-

1444444444444444444444444442444444444444444444444444443

( )

( )

4 5 2 3

,

0 0 0 0

0

0 0

N s z

T sT s

I

T T

é ùê úê úê úê úê úê úê úê úê úê úê ú

+ +ê úê úê úê úë û144444444444444444444444424444444444444444444444443

7 6 5 4 3 27 6 5 4 3 2 1 0,T s z T z s T z s T z s T z s T z s T z s T z s T z


Let now

1 0

1,1 ,0 1,1 1,0 0,1 0,0,

q q

q qs q q q q

L s L s L s

R s z R s R z R s R z R s R

where we have already assumed that q is odd. Applying now, the same techniques with the ones defined above we define

1

1

0 01

,

0 0

0 0 , , 1,2,..., 10

0 0

,

q q rq

rp q k

k rpk k k

rp

rp k

rp q

B s diag L s I

IL s I

B s D s D s k qI

I

B s diag I L s

, ,, ,s zs R s z P s zT s z P s z


3

2 1 0

3 23 ,3

0 0 0 0 0 0 0

, 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

r

s r r r r

L sr

L s L s L s

K I

R s z z z z I sI

M sI

Define the matrices

3 3

3 3 2 1 2 32 3

2 2 3 23

33 2 1

3 3 1 1 31 3

1 1 1 3 13

33 1 1

0

,

0 0 0

0 0 0 0 , 0

0 0 0 0

0 0 0 0

0 0 0 , 0

0 0 0 0

rp

r rr

rr

rr

r rr

rr

rr

B s diag L s I

I L s IL s I

B s D s I DI

I I

I IL s I

B s D s L s I DI

I I

B s

0 6 03 3 1 , ,rrdiag I L s diag I L s


Theorem: ( , ) [ , ]pr prsR s z s z and , [ , ]pqr pqr

zP s z s z are ZC-E.

Let

1 2 0

1 2 0

0,

0

q q q

rp rpz q q q

rp rp

zL s L s L s L s

I sIP s z zB s B s B s B s

I sI

be the companion form of ,sR s z in terms of z .

, ,, ,s zs R s z P s zT s z P s z


3

2 1 0

3 23 ,3

0 0 0 0 0 0 0

, 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

r

s r r r r

L sr

L s L s L s

K I

R s z z z z I sI

M sI

The first companion form (by taking 3 0L s ) of the matrix ,sR s z will be the

following

3 2 1 0

3 3

3 3

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

, 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

r

r r

r

r r

z r r r r

r r r r

r r

r r

r r

K I

I sI

M sI

zL s L s L s L s I zI

P s z I zI I zI

I zI I zI

I zI

I zI

I zI

which is not in symmetric form.


( , ) [ , ]pr prsR s z s z and , [ , ]pqr pqr

zP s z s z are ZC-E.

2

3 3

3

3

( , ) ( , )

0 ( , ) ,

0

r r

s z r

r

M s z N s z

I z I

R s z P s z zI

I


Let the following companion form of ,sR s z and thus of ,T s z given by

2 3

0 1

0, : ,

0

0

pr

z odd even zpr pr

pr

L s zL s IR s z zB s B s R s z

I zI

zI L s zL s

where

0 2 1

1 12 1

even q

odd q q

B s B s B s B s

B s B s B s B s

Theorem: ,zR s z is a self-adjoint matrix pencil which is zero coprime equivalent to

,zP s z and therefore to ,sR s z and in the sequel to ,T s z .

0 1 0,1 0,0 1,1 1,0L s zL s R s R z R s R

1 ,1 ,0 1,1 1,0i i i i i iL s zL s R s R z R s R

, , , , ,z zs sT s z P s z R s z P s z R s z


Define the following companion form

2 3 3 3 ,31

3 1 0 2 3 3 ,3 3

3 ,3 3 0 1

0

, 0

0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

r r r

z r r r r

r r r

r

r

r

r r

r r

r r

r r

r r r

r r

L s zL s I

R s z zB s B s B s B s I zI

zI L s zL s

I

I

M I

I zI

I zI

I zI

zI K I

zI I sI

zI sI


The zero-coprime equivalence that connects the matrix pencils ,zR s z and ,zP s z

will be the following

1

2 3 1 0

2 3

,,

2 3 3

3 3 2 3

0 1

3 3 3

33

3

3

3

,

3

0

0

0 0 0

0 0

0

0 0

0

0 0

0

0

00

z

z

P s zM s z

r

r r

R s z

r r

r r

r

r

r

r

r

I zI

L s zL s L s L s

L s zL s

L s zL s I

I zI L s zL s

L s zL s

I

I

z

I

II

I z

I

1 ,N s z

2-D reduction of polynomial system matrices Let the 2-D polynomial system matrix be the following:

, ,

, R ,, ,

r l r mT s z U s zP s z s z

V s z W s z

or its normalized form

2 2

, , 0 0

, , 0 ,, : R ,

0 0 00 0 0

r l m r m ll

m m

l

T s z U s z

V s z W s z I s zs z s z

I I

I

T UP

V

and let 1 1 2 0,zR s z szE zA sA A

be the reduced matrix pencil of ,s zT according to the algorithm presented in the

previous section.

Theorem: The 2-D polynomial system matrix ,s zP is zero-coprime system

equivalent to a singular model of the form

0

,,

0 0

z

SM

l m

R s zs z

UPV

, ,zs z R s zT

2-D reduction of polynomial system matrices

The polynomial matrix 0 ,A s z is unimodular, and therefore its inverse is a polynomial

matrix. We have also that

1

0 0 0 0 1, ,s z C s A s z B s z s z T T T

0 0

0 0 1

4 5

2 3

0 1

,, :

0 0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0 0 0

0 0 0 0

s

r

r r

r r

r r

r r

r

A s z B sR s z

C s z s z

sI

sI z s z I

I sI

sI z s z I

I sI

sI z s z

T T

T T

T T

T T

0 1, pps z z s z s z T T T T

, ,ss z R s zT


2 3

0 1

0

, : 0

0

r

s r r

r

z s z I

R s z I sI

sI z s z

T T

T T

1

2 30 1

0 12 3

22 3 0 1

2 32 3 0 1

0, 0

0

0 00

0

rr

rr

rr

r r

rr

z s z Is z sI z s z

sII

IsI z s z

I z s z sI

sI s z s z z s zsI

s z s z z s z

T TT T T

T T-T T

T T T T

T T T T


0 0

0 0 1

0 1

, 0,

, :0

0 0

,

S

qq

A s z B ss z

s z C s z s z

s z L s zL s z L s

s 10

10

s

R UP T T U

VV

R

The following transformation

0 0

1

0 0 0

0 1

, , ,,

0

0

, , 0 00 0 , 0

, , 00 0

0 00 0 0 0

0 0

0 ,

0SM s z s z N s z

s z

T s z U s zB s A s z B s

V s z W s z Iz s z

I II I

I

I I

A s z

C s

0

0

PP

T T U

V

is a ZC-S equivalence transformation.


1 1

1 0 1

4 5

2 3

0 1

,, :

0 0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0 0 0

0 0 0 0

z

pr

pr pr

pr pr

pr pr

pr pr

pr

A s z B zR s z

C z L s zL s

zI

zI L s zL s I

I zI

zI L s zL s I

I zI

zI L s zL s

The polynomial matrix 1 ,A s z is unimodular, and therefore its inverse is a polynomial

matrix. We have also that

1

0 1 1 1 1 0 1, ,qs qR s z L s zL s z L s C z A s z B z L s zL s

, ,s zR s z R s z


, , ,ZCSE ZCSE

s SMP s z P s z P s z

The following transformation

1

0 0 1 1 1

0 0 1 0 1

, , , ,

1

1

0 , 0 0 , 0

0 0

0 0 0 0 0 0

0 ,

S SMM s z s z s z N s z

A s z B s B s A s z B s

C s T z sT

A s z

I Cz L s sL z I

I I

s

P P

U U

V V

is a ZC-S equivalence transformation.

Implementation in Mathematica In[1]:=<<LinearAlgebra` In[2]:= f[A1_, s_] := Module[{n, r, T, B, i, k, EE, AA, j, tmp}, n = Max[Exponent[A1, s]]; r = Length[A1]; If[Mod[n, 2] == 0, T[i_] := Coefficient[A1, s, i]; T[n + 1] := ZeroMatrix[r]; n = n + 1, T[i_] := Coefficient[A1, s, i]];

B[0] := BlockMatrix[Table[If[i == j && j n, IdentityMatrix[r], If[i j, ZeroMatrix[ r], -T[0]]], {i, 1, n}, {j, 1, n}]]; B[n] := BlockMatrix[Table[If[i == j == 1, T[n], If[ i == j, IdentityMatrix[r], ZeroMatrix[r]]], {i, 1, n}, {j, 1, n}]]; MkI[ll_ , mm_] := Module[{i, j, mat}, mat = mm; For[i = ll[[1, 1]], i ll[[1, 2]],

For[j = ll[[2, 1]], j ll[[2, 2]], mat[[i, j]] = If[Abs[j - i] == ll[[1, 2]] - ll[[1, 1]] + 1, 1, 0]; j++]; i++]; Return[mat]]; MkZ[ll_ , mm_] := Module[{i, j, mat}, mat = mm;

For[i = ll[[1, 1]], i ll[[1, 2]], For[j = ll[[2, 1]], j ll[[2, 2]], mat[[i, j]] = 0; j++]; i++]; Return[mat]]; B[k_] := Block[{tmp, i, j, n1 = n, r1 = r}, tmp = BlockMatrix[Table[If[i == j == n1 - k, -T[k], If[i == j, IdentityMatrix[r1], ZeroMatrix[r1]]], {i, 1, n1}, {j, 1, n1}]]; If[r1 > 1, tmp = MkZ[{{(n1 - k)r1 +1, (n1 - k + 1)r1}, {(n1 - k)r1 + 1, (n1 - k + 1)r1}}, tmp], tmp[[{(n1 - k)r1 + 1, (n1 - k + 1)r1}, {(n1 - k)r1 +1, (n1 - k + 1)r1}]] = 0]; If[r1 > 1, tmp = MkI[{{(n1 - k - 1)r1 + 1, (n1 - k)r1}, {(n1 - k)r1 + 1, (n1 - k + 1)r1}}, tmp], tmp[[{(n1 - k - 1)r1 + 1, (n1 - k) r1}, {(n1 - k)r1 + 1, (n1 - k + 1)r1}]] = 1]; If[r1 > 1, tmp = MkI[{{(n1 - k)r1 + 1, (n1 - k + 1)r1}, {(n1 - k - 1)r1 + 1, (n1 - k)r1}}, tmp], tmp[[{(n1 - k)r1 + 1, (n1 - k + 1)r1}, {(n1 - k - 1)r1 + 1, (n1 - k)r1}]] = 1]; tmp]; EE = B[n];

For[i = n - 2, i 1, EE = EE.Inverse[B[i]]; i = i - 2]; AA = B[0]; For[i = 2, i n - 1, AA = AA.B[i]; i = i + 2]; Return[s EE - AA]]

Implementation in Mathematica Example: First we define the matrix 2 21A Ks Mz

[ ] : {{ }}

[ ]: {{ ^ 2 ^ 2}}K s M z

In 3

Out 3

A1 Ks 2̂ Mz 2̂

Then we reduce the matrix 1A to a matrix pencil in terms of s [ ] : [ , ]

[ ] : {{ , 1,0},{ 1,0, },{0, , ^ 2}}K s s M z In 4

Out 4

f A1 s

and finally we reduce the last result to a matrix pencil in terms of z [ ] : [%, ]In 5 f z [ ] :Out 5 {{0,0,0, 1,0,0,0,0,0},{0,0,0,0, 1,0,0,0,0}, {0,0, ,0,0, 1,0,0,0},{ 1,0,0,0,0,0, ,0,0},M z {0, 1,0,0,0,0,0, ,0},{0,0, 1,0,0,0,0,0, },z z {0,0,0, ,0,0, , 1,0},{0,0,0,0, ,0, 1,0, },z K z s {0,0,0,0,0, ,0, ,0}}z s

Implementation in Mathematica Note that the above reduction function []f may also be used for polynomial matrices with more than two variables.

[ ] : {{ N w^2}};

[ ]: [ , ];

[ ] : [%, ];

[ ] : [%, ]//

In 7

In 8

In 9

In 10

A1 Ks 2̂ Mz 2̂

f A1 s

f z

f w Mat r i xFor m

Conclusions A two-stage algorithm, easily implementable in a computer symbolic environment, has been provided for the reduction of a 2-D symmetric polynomial matrix to a zero coprime equivalent 2-D symmetric matrix pencil.

The results has also been adapted to 2-D system matrices.

Advantage.

We can use existing robust numerical algorithms for 2-D matrix pencils in order to compute structural invariants of 2-D symmetric polynomial matrices

Disadvantage.

The size of the matrices that we use.

An implementation of this algorithm in the package MATHEMATICA accompanied with one example is given.

Further research

Reduction of symmetric and positive definite polynomial matrices.

Use of other matrix pencil reduction methods (Higham et.al.).

New numerical techniques for investigating structural invariants of 2-D symmetric matrix pencils.

Infinite elementary divisor structure ?

Illustrative ExampleExample: Consider the system described by the following partial differential equations :

2 2

2 2

, ,0

u x y u x yK M

x y

By replacing

2

2 2

2

2 2

, , ,

, 2 , 2 , ,

, , ,

, , 2 2 , ,

u x y u x k y u x y

x k

u x y u x k y u x k y u x y

x ku x y u x y h u x y

y h

u x y u x y h u x y h u x y

y h

Illustrative Examplewe get the following discretized system

2 2

2 2

2 , 2 , , , 2 2 , ,0

2 , 2 , , , 2 2 , , 0

u x k y u x k y u x y u x y h u x y h u x yK M

k h

Kh u x k y u x k y u x y Mk u x y h u x y h u x y

The 2-D polynomial matrix that corresponds to this system is the following :

2 1 0

2 2 2 2

2 2 2 2 2 2

, 2 1 2 1

2 2 1

T s T s T s

T s z Kh s s Mk z z

Mk z Mk z Mk Kh s s

Thus if the original polynomial matrix 2 2Ks Mz is symmetric then the polynomial matrix ,T s z will also be symmetric.

Illustrative Example

2 1 0

2

2 2 2 2

2

2

2 2 2 2 2

0

, : 0

0 2 1 2

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 2 0 2

r

z r r

r

r

r r

r

L z L z L z

Mk I

R s z I zI

zI Mk Kh s s Mk z

Mk I

s s I zI

Kh Kh zI Mk Kh Mk z

Illustrative Example

2

2

2 2 2 2

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

, 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 2 2

r

r

r

r r

s r r

r r

r r

r r r

r r

I

I

Kh I

I sI

R s z I sI

I sI

sI Mk I

sI I zI

sI zI Mk Kh Mk z Kh s

Motivation – 1-D polynomial matrices

we get

Consider the homogeneous system

1 11 1 0

11 1 0

0

0, : /

n nn n

n nn n

T

T x t T x t T x t T x t

T T T T x t d dt

Then by defining the following variables

11

1 1 22 1 1

1 11

1

,

.....

1

nn

nn

nn n

N

x t x t x t

x t x t x t x tx t

x tx t x t x t

11 1 1 2 0 1 0n n n nT x t T x t T x t T x t

or equivalently


is known as the first companion form of T(ρ).

( ) ( )

( ) ( )

( ) ( )

( )

( )

( )

11 2 0

111

1 11

0 0

0 00

0

0 00 0

nn n n n

npp n

pp

x tT T T T x t

x tII x t

x tII x t

- -

--

é ù - - -é ù é ùé ùê úê ú ê úê úê úê ú ê úê úê úê ú ê úê úê ú=ê ú ê úê úê úê ú ê úê úê úê ú ê úê úê úê ú ê úê úë ûê ú ë ûë ûë û

L L

LO M

MM O O MM O O M

LL

and the following matrix pencil

1 2 0

0

0

n n n

p p

p p

T T T T

I IE A

I I

r

rr

r


L

O M

M O O

L


in the sense that the compound matrices

( ){

( )

( )

( )

11 2 0

20 0

0

0 0 0

nn n n pp

np p p

p p p

M E A N

T T T T II T

I I IT

I I Ir r r

r rr

r rr

r

-- -

-

-

é ù+é ùé ùé ù ê úê úê úê ú ê úê úê úê ú - ê úê úê úê ú = = ê úê úê úê ú ê úê úê úê ú ê úê úê úê ú ê úê úê úê ú -ë û ê úë û ë ûë û

L

O M

M M M O O M

L14444444444444244444444444443144424443

Note that the following extended unimodular equivalent transformation connects the polynomial matrix T(ρ) and the respective pencil ρE-A.

; T

M E AN

do not lose rank in C.

Conclusion. Since, T(ρ) and ρE-A are e.u.e. they possess the same finite elementary divisor structure.

Motivation – 1-D symmetric polynomial matrices

Disandvantages. Numerical methods that ignore the special structure of the

polynomial matrix T(ρ) (like the ones above) often destroy these qualitatively important spectral symmetries, sometimes even to the point of producing physically meaningless or uninterpretable results.

Storage and computational cost are reduced if a method that exploits symmetry is applied i.e. The solution of Ax=B, with A symmetric, via a symmetric banded solver uses O(n) storage and O(n) flops, while using LU methods that not exploits symmetry uses O(n2) storage and O(n3) flops.

Consider the symmetric polynomial matrix

11 1 0: , n n T

n n i iT T T T T T T

and the respective e.u.e. matrix pencil

1 2 0

:0

0

n n n

p p

p p

T T T T

I IE A

I I

r

rr

r


L

O M

M O O

L

Motivation – 1-D symmetric polynomial matrices

we get the following homogeneous system

Consider the homogeneous system 5 4 3 2 1

5 4 3 2 1 0 0, Ti iT x t T x t T x t T x t T x t T x t T T

Then by defining the following variables

1

1 352 2 3 4 5

4 4 52 3 3 4

313 1

2 4 52

4 4 5 4 5 5 12

5 3

,

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

y t x t

y t Iy t T T x t T T x ty t T TT T y t y ty t I

y t x t y ty t T T

y t T T x t T T y t y t I

y t x t y t

5

4

3

2

1

N

x t

x t

x t

x t

x t


is e.u.e. to ρE-A.

where the symmetric matrix pencil

4 5

2 31 1

0 1

0 0 0

0 0 0

0 0

0 0 0

0 0 0

T T I

I I

I T T IE A

I I

I T T

r

r

r rr

r

r r

+ -é ùê úê ú-ê úê ú+ -- = ê úê ú

-ê úê úê ú+ê úë û

4 5 5

4

2 3 3

2

0 1 1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

T T I y t

I I y t

I T T I y t

I I y t

I T T y t


( )

4 5 3 2 1 0

24 5 3 4 5 2 3 1

0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

0

0 0 0

0 0 0

0 0

0 0 0

0 00

E AM

I I

I I

I I

I

T T T T T

I

I

I

I

T

T T TT T TT Trr

r

r r r

r

r

rr

r

-

+é ùé ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úê ú+ + +

-

-

-

ê úê úëû+ ûë-

144444444444 2144444444444444444444424444444444444444444443

( ) ( )

1 1

24 5 2 3 0 1

4 5

2 3

0 1

0

0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0

0 0

0 0

00 0

0

0

0 0

0

0

00

E A

I I

I I

T T T T T T

I

I I

T T

I I

T T

I I

T TI

r

r

r

r r r r

r

r

r

r

r

rr

r

-

-

-

=

é ùê úê ú-ê úê ú

= =ê úê ú

-ê úê úê ú+ + +ê úë û

+é ùê úê ú-ê úê ú+= ê úê ú

-ê úê úê ú+ê úë û

44 44444444444443

144444444444444444442444444444 3

( )

( )

4 5

4 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0

0

0

0 0 0

N

I

T T

I

T T

I

r

r

r r

é ùê úê ú+ê úê úê úê úê ú+ê úê úê úë û4444444444 144444444444424444444444443

Documents

An equivalent reduction of a 2-D symmetric polynomial matrix N. P. Karampetakis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki