vol5_pp67-103. 355KB Jun 04 2011 12:06:14 AM

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The Electronic Journal of Linear Algebra.

A publication of the International Linear Algebra Society. Volume 5, pp. 67-103, May 1999.

ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

STRUCTURED JORDAN CANONICAL FORMS FOR STRUCTURED

MATRICES THATARE HERMITIAN, SKEW HERMITIAN OR

UNITARY WITH RESPECTTO INDEFINITE INNERPRODUCTS

VOLKER MEHRMANNy

AND HONGGUO XU

y Abstract. For inner products dened by a symmetric indenite matrix

p;q, canonical forms for real or complex p;q-Hermitian matrices, p;q-skew Hermitian matrices and p;q-unitary matrices are studied under equivalence transformations which keep the class invariant.

Keywords.structured eigenvalue problems, Lie group, Lie algebra, Jordan algebra AMSsubject classications.15A21, 65F15

1. Introduction. In several recent papers [1, 8, 10, 9, 6] the topic of canonical

forms for structured matrices and pencils associated with classical Lie groups, Lie algebras and Jordan algebras has been studied. The motivation for these analyses is the development of new structure preserving numerical methods for the solution of the eigenvalue problem for matrices in these classes. The main motivation is to use equivalence transformations that preserve the algebraic structures, i.e., for example the symmetry in the spectrum in nite arithmetic. This means that the transfor-mation matrices are restricted to be from the associated Lie groups only. If such structure preserving methods can be constructed, then this usually leads to a reduc-tion in complexity and at the same time it avoids that in nite arithmetic physically meaningless results are obtained. Often one also has a better perturbation and error analysis, see for example [2, 3, 4]. The latter is obtained in particular if one uses unitary transformations which are at the same time in the associated Lie group, since then the methods, usually, are also numerically backwards stable. However, for nu-merical computations we need to know the proper condensed forms within the given structures, usually called structured Schur like forms, that the numerical methods can possibly generate, and from which the eigenvalues and eigenstructures can be easily read o. The structured Jordan like canonical forms that we describe here are the simplest versions of such condensed forms, although they need nonunitary transforma-tions. Hence these Jordan like forms will be the fundamental theory for studying the proper structured Schur like forms and therefore for developing numerical methods.

The invariants under similarity transformations have been classied already for quite a while [5, 11]. There, for physical applications the canonical forms are restricted to be classical Jordan forms. So the transformation matrices are not in the associated Lie groups. Such canonical forms are not what we are interested in. Hence it is necessary to convert these forms to the desired forms.

A complete analysis for the case of Hamiltonian, skew Hamiltonian and symplectic matrices, i.e., matrices that are Hermitian, skew Hermitian and unitary with respect to an indenite scalar product given by a skew symmetric matrix, has recently been given in [8]. In this paper we now derive analogous results for the matrices that are Hermitian, skew Hermitian and unitary with respect to an inner product dened via

Received by the editors on 13 October 1998. Accepted for publication on 30 April 1999. Handling Editor: Daniel Hershkowitz.

yFakultat fur Mathematik, TU Chemnitz, D-09107 Chemnitz, FR Germany. This work has been supported byDeutsche Forschungsgemeinschaftwithin SFB393 `Numerische Simulation auf massiv parallelen Rechnern'

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ELA

68 Structured Jordan Canonical Forms

the indenite symmetric matrix p;q :=

I

p 0

0 ,I q

, whereI k is the

kkidentity

matrix. We consider the following classes of matrices.

Definition 1.1. LetR andC denote the real and complex eld, respectively.

A matrix C 2C

(p+q)(p+q) is called

p;q-Hermitian if C

p;q = ( C

p;q) H.

C is called

p;q-symmetric if it is p;q-Hermitian and real.

A matrix C2C

(p+q)(p+q)

is called p;q-skew Hermitian if C

p;q = ,(C

p;q) H.

C is

calledp;q-skew symmetric if it is p;q-skew Hermitian and real.

A matrix G 2 C

(p+q)(p+q) is called

p;q-unitary if G

H p;q

G =

p;q. It is called

p;q-orthogonal if it isp;q-unitary and real. Note that the p;q-Hermitian matrices

form a Jordan algebra, the p;q-skew Hermitian matrices from a Lie algebra, and the

p;q-unitary matrices form a Lie group. The algebras and group are invariant under

similarity transformations with p;q-unitary matrices. Proposition1.2.

1. If C is

p;q-Hermitian and G is

p;q-unitary then G

,1

CG is

p;q-Hermitian.

2. If C is

p;q-skew Hermitian and G is

p;q-unitary then G

,1

CG is

p;q-skew

Her-mitian. 3. If G

1 and G

2 arep;q-unitary then G

1 G

2 is alsop;q-unitary.

Similar to the approach for Hamiltonian and symplectic matrices in [8] we derive structured Jordan canonical forms for these classes of matrices. But dierent from the case of Hamiltonian and symplectic matrices and pencils, for matrices that are p;q-Hermitian, skew Hermitian or unitary, it is dicult to derive the appropriate

structured Schur like forms with similarity transformations that are both unitary and p;q-unitary, since this class has only a very small dimension. Currently the best that

one can do in this respect are the shbone like forms of [1]. As mentioned above the approach that we present here will be taken as the rst step for the structured Schur-like forms. To make the idea more clear let us consider the case of p;q-Hermitian

matrices. The discussion for the other cases is similar. There are many dierent approaches that one can take to derive canonical and condensed forms for such ma-trices. A very simple approach to obtain a canonical form is the idea to express the p;q-Hermitian matrix

C as an Hermitian pencil p;q

,

p;q

C. Using congruence

transformationsU H(

p;q ,

p;q

C)U, we obtain a canonical form via classical results,

see e.g., [7, 12, 13, 5]. In view of our goals, however, this is not quite what we want, since in general these forms do not give thatU

H p;q

U =

p;q, hence they do not lead

directly to the desired structured form. Clearly, however, the characteristic quantities that we obtain from this canonical form will have to appear in our canonical form, too.

The outline of the paper is as follows: We will present some basic preliminary results and some notations in Section 2 and then present structured canonical forms for p;q-Hermitian matrices and p;q-skew Hermitian matrices under p;q-unitary

similarity transformations in Section 3 and Section 4, respectively. By combining the Cayley transformation and the structured canonical forms for p;q-skew Hermitian

matrices we will then derive the structured canonical forms for p;q-unitary matrices

in Section 5. All canonical forms are represented both for real and complex matrices. For comparisons and derivations we also list the already known classical canonical forms.

The theorems for the main results are listed in Table 1.1. HereC andRrepresent

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Volker Mehrmann and Hongguo Xu 69

p;q-Hermitian p;q-skew Hermitian p;q-unitary

C

& J Theorem 3.1 Theorem 4.1 Theorem 5.9

R

& J Theorem 3.2 Theorem 4.2 Theorem 5.10

C

& U Theorem 3.3 Theorem 4.3 Theorem 5.11

R

& U Theorem 3.6 Theorem 4.4 Theorem 5.12

Table1.1 Mainresults

transformations, respectively.

2. Preliminaries.

In this section we introduce the notation and give some pre-liminary results that are needed for the canonical forms. Our construction of struc-tured Jordan froms will be based on the combination of dierent blocks of the classical, unstructured Jordan form. Let us recall some facts from the classical theory.

Let (A) denote the spectrum of a matrixA. We begin with a well-known fact

on the relationship between left and right invariant subspaces, which follows clearly from the Jordan canonical form.

Proposition 2.1. Let the columns of U span the left invariant subspace of a

square matrix Acorresponding to 1

2(A) and let the columns ofV span the right

invariant subspace corresponding to 2

2 (A). If 1

6

= 2 then

U H

V = 0 and if

1=

2 thendet( U

H V)6= 0:

Let

N r:=

2 6 6 6 6 4

0 1 ... ...

... 1 0

3 7 7 7 7 5

be anrrnilpotent Jordan block. Dene accordingly

P r:=

2 6 6 4

,1

(,1) 2

(,1) r

3 7 7 5

; P^ r:=

2 6 6 4

1

3 7 7 5

rr :

For any given nilpotent matrix

N= diag(N r1

;:::;N rs)

we set

P

N := diag( P

r1 ;:::;P

rs) ; P^

N := diag( ^ P r1

;:::;P^ rs)

:

Then these matrices have the following easily veried properties.

Proposition2.2.

i)P H r =

P ,1 r = (

,1) r,1

P r;

ii) P ,1 N

N H

P N=

,N;

iii) ^P N= ^

P H N = ^

P ,1 N ;

iv) ^P ,1 N

N H^

P N =

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For matricesAandB, AB = [a ij

B] denotes the Kronecker product and for a ttmatrixZ witht= 1, or 2 we set

N r(

Z) :=I r

Z+N r

I t

; N(Z) :=IZ+NI t

:

Also set

1;1=

1 0 0 ,1

; J

1=

0 1

,1 0

:

From [5], or from a direct derivation as done in [8] we have the following properties.

Proposition2.3. LetC be a complex square matrix and letbe an eigenvalue

ofC with associated Jordan structureN(). If :=

Re Im

,Im Re

, then we have the following results.

I.a If C is complex

p;q-Hermitian, then there exists a nonsingular matrix

U such

that

i) if Im6= 0, then

U H

p;q U =

0 ^P N

^

P H

N 0

; CU =U

N() 0

0 N()

;

ii) ifis real, then U

H p;q

U = diag( 1^

P r1

:::; s^

P rs)

; CU =UN();

where all i=

1. I.b If C is real

p;q-symmetric, then there exists a real nonsingular matrix

U such

that

i) if Im6= 0, then U

T p;q

U = ^P N

1;1

; CU =UN();

ii) ifis real, then U

T p;q

U = diag( 1^

P r1

:::; s^

P rs)

; CU =UN();

where all i=

1. II.a If C is complex

p;q-skew Hermitian, then there exists a nonsingular matrix U

such that

i) if Re6= 0, then

U H

p;q U =

0 P N P

H

N 0

; CU =U

N() 0

0 N(,)

;

ii) ifRe= 0, then U

H p;q

U = diag( 1

P r

1 :::;

s P

r s)

; CU =UN();

where k=

ifor evenr k and

k =

1 if odd r k. II.b If C is real

p;q-skew symmetric, then there exists a real nonsingular matrix U

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i) if ReIm6= 0, then

U T

p;q U =

0 P

N

1;1 P

T N

1;1 0

; CU =U

N() 0

0 N(,)

;

ii) ifRe6= 0, Im= 0 then

U T

p;q U =

0 P N P

T

N 0

; CU =U

N() 0

0 N(,)

;

iii) if 6= 0, Re= 0, then U

T p;q

U = diag(P r1

1 ;:::;P

rs

s)

; CU =UN((Im)J 1)

;

where k =

k I

2 for odd r

k and k = (Im

k) J

1 for even r

k, and

k is as in the

complex case. iv) if= 0, then

U T

p;q

U = diag

1

P 2u1+1

;:::; a

P 2ua+1

;

0 P 2v1 P

T 2v

1 0

;:::;

0 P 2vb P

T 2v

b 0

; CU =UN :=Udiag(N

2u 1

+1 ;:::;N

2u a

+1 ;N

2v 1

;N 2v

1 ;:::;N

2v b

;N 2v

b) :

Note that the parameters

k are invariant in the sense that for each group of Jordan

blocks with the same size corresponding tothe numbers of 1, ,1, or i, ,i, of the

corresponding

k are uniquely determined. For this reason we denote the complete set

of parameters by Ind() =f 1

;:::; s

gand call this the structure inertia index. Note

that for each Jordan block there is a unique corresponding structure inertia index. Some obvious facts on the symmetry of the eigenvalues follow directly from Propo-sition 2.3.

Proposition 2.4. Let be an eigenvalue of a square matrixC. Then we have

the following properties.

I. If C is

p;q-Hermitian (both complex and real) and Im

6= 0 then is also an

eigenvalue of C with the same Jordan structure as. II.aIf C is complex

p;q-skew Hermitian and

is not purely imaginary, then, is

also an eigenvalue of C with the same Jordan structure as. II.bIf Cis real

p;q-skew symmetric and

is not purely imaginary, then ,,,,

are eigenvalues ofC with the same Jordan structures as. If= 0 is an eigenvalue,

then the number of each even sized corresponding Jordan blocks must be even. We will frequently use transformations with

r= p

2 2

I

r ,I

r I

r

;

(2.1)

for which we have

H r

0 I r I

r 0

r=

I

r 0

0 ,I r

:

(2.2) ForA2C

nna simple calculation yields

,1 n

A 0

0 A

H

n = "

AA H 2

, AA

H 2 ,

AA H

# :

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A variation of ris

^r= p

2 2

2 4

I

r 0

,I r

0 p

2 0

I

r 0

I r

3 5 :

(2.4)

We will also need, in the following, the symmetric and skew symmetric part of Jordan blocks

N + r = 12(

N r+

N T r)

; N

, r = 12(

N r

,N T r)

;

and for attmatrixZ witht= 1, 2, we set N

+ r(

Z) =I r

Z+N + r

I t

; N

, r (

Z) =I r

Z+N , r

I t

:

Similarly we denote

N +

= 12(N+N T)

; N

,

= 12(N,N T)

;

and forZ ofttwitht= 1, 2, N

+(

Z) =I Z+N +

I t

; N

,(

Z) =IZ+N ,

I t

:

Finally the symbole

k represents the

k-th unit vector.

With these notations and results in hand in the next two sections we will give the canonical forms for p;q-Hermitian and p;q-skew Hermitian matrices.

3.

p;q

-Hermitian matrices.

In this section we derive structured Jordan

canon-ical forms for p;q-Hermitian matrices. We will always consider two forms, a

struc-tured canonical form where the transformation matrices are not necessarily p;q

-unitary and a structured canonical form under p;q-unitary matrices. Also we will

always consider two cases, the complex p;q-Hermitian and the real p;q-symmetric

matrices.

Theorem 3.1. LetC be a complex

p;q-Hermitian matrix with pairwise dierent

real eigenvalues 1

;:::;

and pairwise dierent eigenvalues

1 ;:::;

, with positive

imaginary parts. Then there exists a nonsingular matrix U such that U

,1

CU = diag(R + c

;R , c

;R r)

;

where the blocks are

R +

c = diag( H

1(

1) ;:::;H

(

))

; R

,

c = diag( H

1(

1) ;:::;H

(

)) ; R

r= diag( M

1(

1) ;:::;M

(

)) ;

with substructures

H k(

k) =

k

I+H k

; H

k(

k) =

k I+H

k

; H

k = diag( N

p k ;1

;:::;N p

k ;s k)

;

for k= 1;:::;, and M

k(

k) =

k I+M

k

; M

k= diag( N

qk ;1 ;:::N

qk ;t k)

;

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The matrixU satises

U H

p;q U =

2 4

0 W

c 0

W H

c 0 0

0 0 W

r 3 5 ;

(3.1) with W

c = diag( ^ P H

1 ;:::;P^

H ) and

W

r = diag( W

r 1

;:::;W r

), where for

k= 1;:::;

we have ^P H

k = diag( ^ P p

k ;1 ;:::;P^

p k ;s

k) and for

k= 1;:::; and Ind( k) =

f k ;1,

:::,

k ;t k

gwe have W r

k = diag(

k ;1^ P q

k ;1 ;:::;

k ;t k^

P q

k ;t k).

Proof. For each nonreal eigenvalue

k with the corresponding Jordan structure H

k(

k), by I.a, i) of Proposition 2.3, we can choose a matrix U

k such that U

H k

p;q U

k =

0 P^ H

k

^

P H H

k 0

; CU

k= U

kdiag( H

k(

k) ;H

k(

k)) :

(3.2) PartitionU

k= [ U

k ;1 ;U

k ;2], where U

k ;1, U

k ;2 have the same size and set U

c= [ U

1;1 ;:::;U

;1; U

1;2 ;:::;U

;2] = [ U

c 1;

U c 2]

:

Note that by the symmetry of C, the columns of U c 1,

p;q U

c 2 and

U c 2,

p;q U

c 1 form

bases of the right and left invariant subspaces corresponding to the two disjoint sets of eigenvalues f

1 ;:::;

g and f 1

;:::;

g, respectively. By Proposition 2.1 and

(3.2) we have

U H c

p;q U

c =

0 W

c W

H

c 0

; CU =Udiag(R + c

;R , c)

;

withW c,

R + c and

R ,

c as asserted.

For each real eigenvalue

k with the corresponding Jordan structure M

k(

k), by

I.a, ii) of Proposition 2.3 we can choose a matrixV

k such that V

H

k

p;q V

k= diag(

k ;1^ P

qk ;1 ;:::;

k ;tk^ P

qk ;t k)

;

where Ind( k) =

f k ;1

;:::; k ;t

k

gandCV k =

V k

M k(

k). Set

U r= [

V 1

;:::;V ], then

by Proposition 2.1 we have

U H r

p;q U

r= W

r ; CU

r= U

r R

r ;

where W r and

R

r are of the asserted forms and with

U = [U c

;U

r] the result follows

from Proposition 2.1.

Similarly for real p;q-symmetric matrices by employing I.b of Proposition 2.3 we

have the following forms.

Theorem 3.2. Let C be a real

p;q-symmetric matrix with pairwise dierent

real eigenvalues 1

;:::;

1 ;:::;

, with positive

imaginary parts.

Then there exists a real full rank matrix U such that U

,1

CU= diag(R c

;R r)

;

where

R

c= diag( H

1(1) ;:::;H

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and fork= 1;:::;the subblocks areH

k(k) = diag( N

pk ;1(k) ;:::;N

pk ;s k(

k)) with

k =

Re

k Im

k ,Im

k Re

k

. The other diagonal block is

R

r= diag( M

1(

1) ;:::;M

(

)) ;

where for k= 1;:::; the subblocks are M k(

k) =

k

I+M k with

M

k = diag( N

qk ;1, :::,N

qk ;t

k). The matrix

U has the form U

T p;q

U =

W

c 0

0 W r

;

(3.3) whereW

c = diag( ^ P

H1

1;1 ;:::;P^

H

1;1), W

r= diag( W

r 1

;:::;W r

), and where for k= 1;:::;we have ^P

Hk = diag( ^ P

pk ;1 ;:::;P^

pk ;s

k) and for Ind(

k) = f

k ;1 ;:::;

k ;tk g

andk= 1;:::; we haveW r

k = diag(

k ;1^ P q

k ;1 ;:::;

k ;t k

^

P q

k ;t k).

The canonical forms in Theorems 3.1 and 3.2 are just the results of [5] in matrix form. They are just the classical Jordan canonical forms, but the transformation matrices are constructed in such a way that they satisfy the relationship (3.1) and (3.3), respectively, associated with p;q. This is not quite what we want, since we

wish to have that the transformation matrix is p;q-unitary. The following results

give the structured canonical forms under p;q-unitary transformations. Theorem 3.3. Let C be a

p;q-Hermitian matrix with pairwise dierent real

eigenvalues 1

;:::;

1 ;:::;

, with positive

imaginary parts. Then there exists ap;q-unitary matrix

U such that

U ,1

CU = 2 6 6 4

R c

T c R

+ r

T r ,T

H c

R c ,T

H r

R , r

3 7 7 5 :

(3.4)

For the blocks we have the following substructures.

i) The blocks with index c, associated with the nonreal eigenvalues, are R

c= diag( R

c 1

;:::;R c )

; T

c= diag( T

c 1

;:::;T c )

;

where for k= 1;:::;we have R

c

k= diag( N

+ p

k ;1(Re

k) ;:::;N

+ p

k ;s k(Re

k))

; T

c k =

,diag(N , p

k ;1( iIm

k) ;:::;N

, p

k ;s k

(iIm k))

:

ii) The blocks with indexr, associated with the real eigenvalues are R

+

r = diag( C

1 ;:::;C

)

; R

,

r = diag( D

1 ;:::;D

)

; T

r= diag( F

1 ;:::;F

) :

Fork= 1;:::; these have the substructures C

k = diag( C

e k

;C + k

;C , k)

; D

k = diag( D

e k

;D + k

;D , k)

; F

k= diag( F

e k

;F + k

;F , k )

;

where

C e

k = diag( N

+ q

k ;1(

k) + 12

k ;1 e

q k ;1

e H q

k ;1 ;:::;N

+ q

k ;t k

( k) + 12

k ;t

k e

q k ;t

k e

H q

k ;t k

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Dek= diag(N+

qk ;1(k) ,

1 2k;1eq

k ;1eHq

k ;1;:::;N +

qk ;t k(k)

,

1

2k;tkeqk ;t keHq

k ;t k);

Fek= diag(,N ,

qk ;1 + 12k; 1eq

k ;1eHq k ;1;:::;

,N ,

qk ;t

k + 12k;t keqk ;t

keHq k ;t

k);

C+

k = diag

"

N+

uk ;1(k) p

2 2 eu

k ;1 p

2

2 eHuk ;1 k #

;:::;

"

N+

uk ;w k

(k) p 2 2 eu

k ;w k p

2 2 eHuk ;w

k

# !

; D+

k = diag(N+

uk ;1(k);:::;N +

uk ;w k(k));

F+

k = diag

" ,N

,

uk ;1 p

2 2 eHuk ;1

#

;:::;

" ,N

,

uk ;w k p

2 2 eHuk ;w

k #!

; C,

k = diag(N+

vk ;1(k);:::;N +

vk ;z k

(k));

D,

k = diag

"

k , p

2 2 eHvk ;1 ,

p 2 2 ev

k ;1 N +

vk ;1(k) #

;:::;

"

k , p

2 2 eHv

k ;z k ,

p 2 2 ev

k ;z

k N

+

vk ;z k

(k)

#!

;

F,

k = diag([

p

2 2 evk ;1;

,N ,

vk ;1];:::;[ p

2 2 evk ;z

k; ,N

,

vk ;z k]): Hereeach nonreal k (k) has sk Jordan blocks of sizes pk;

1;:::;pk;s k

andeach realeigenvalue k has

a)tkevensizedJordanblocksofsizes2qk;

1;:::;2qk;t k

andthecorrespondingstructure inertia indicesk;

1;:::;k;t k

;

b)wk oddsizedJordanblocks ofsizes2uk;

1+ 1;:::;2uk;w k+ 1

, corresponding tothe structureinertia index 1;

c) zk odd sizedJordan blocks of sizes 2vk;

1+ 1;:::;2vk;z k+ 1

, corresponding to the structureinertia index ,1.

Proof. Letm := P

i=1 Ps

i

j=1pij. For

1;:::;, by Theorem 3.1 there exists a

matrixUc such that

^

UHcp;qU^c=

0 Wc

W_cH ₀

; CU^c= ^Uc

R+

c 0

0 R,

c

:

Note W_cH_W_c ₌_W2

c =Im. Setting ~Uc := ^Ucdiag(Im;Wc) then using the form ofR ,

c

and Proposition 2.2 we have ~

UHcp;qU~c=

0 Im

Im 0

; CU~c = ~Uc

R+

c 0

0 (R+

c)H

:

Now letUc:= ~Ucm, where mis dened in (2.1). By (2.2) and (2.3) and the special

form ofR+

c we then get

UHcp;qUc = m;m; CU =U

Rc Tc

,TcH Rc

;

(3.5)

whereRc andTcare in the asserted forms.

For real eigenvalues1;:::; the situation is relatively complicated. In this case

we have to transform the Jordan blocks one by one. Let be a real eigenvalue ofC

and_^ Nr() be a Jordan block. Following from Proposition 2.3 there exists a matrix

U such that

^

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where=1. Whenris even letU := ^Udiag(I

2 then one can verify that U

k be a real eigenvalue with associated even sized Jordan blocks of sizes 2 q

k ;1, :::, 2q

k ;tk, and associated odd sized Jordan blocks of sizes 2 u

k ;zk+ 1 corresponding to the structure inertia indices 1 and ,1

respectively. By Proposition 2.3 there exists a matrix ^U

k such that

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where the columns of V_ek;j;W_ek;j are qk;j, the columns of V+

k;j, W+

k;j, V,

k;j, W,

k;j are

uk;j+ 1,uk;j,vk;j,vk+1+ 1, respectively. Set

Vk = [Vek;1;:::;Vek;t k; V

+

k;1;:::;V +

k;wk; V ,

k;1;:::;V ,

k;zk];

Wk = [Wek;1;:::;Wek;t k; W

+

k;1;:::;W +

k;wk; W ,

k;1;:::;W ,

k;zk]

andUk = [Vk;Wk]. Then by employing (3.6) { (3.8) we have

U_Hkp;qUk =

Ink ;1 0

0 ,In k ;2

; CUk=Uk

Ck Fk

,FkH Dk

;

whereCk,Fk,Dk are as asserted,nk;1=wk+ Pt

k

l=1qk;l+ Pw

k

l=1uk;l+ Pz

k

l=1vk;land

nk;2=zk+ Pt

k

l=1qk;l+ Pw

k

l=1uk;l+ Pz

k

l=1vk;l. Setn 1=

P

k=1nk; 1, n2=

P

k=1nk; 2.

Then with

Vr= [V1;:::;V]; Wr= [W1;:::;W];

andUr= [Vr;Wr] we have

UHrp;qUr=

In1 0

0 ,In 2

; CUr=Ur

R+

r Tr

,TrH R ,

r

;

(3.9)

Finally setU = [Vc;Vr; Wc;Wr], then by Proposition 2.1 and by above

construc-tion we have

UHp;qU =

Im+n

1 0

0 ,Im +n

2

:

SinceUis nonsingular it follows thatUHp;qUis congruent to p;qand hencem+n 1=

p,m+n2=qand

UHp;qU = p;q. Equation (3.4) then follows from (3.5) and (3.9). Remark 3.4. The dierence between the structured canonical forms of The-orems 3.1 and 3.3 is that in order to get a p;q-unitary transformation matrix we

need to rene further and combine dierent blocks together. This leads to a loss in structure in the Jordan canonical form, which becomes more complicated, but shows that the classical Jordan canonical form somehow obscures the extra structure in the chains of root vectors.

Remark 3.5. By the structured Jordan form we immediately obtain the following relationships

p=X

k=1

sk X

j=1

pk;j+X

k=1

(wk+ t

k X

j=1

qk;j+ w

k X

j=1

uk;j+ z

k X

j=1

vk;j);

q=X

k=1

sk X

j=1

pk;j+X

k=1

(zk+ t

k X

j=1

qk;j+ w

k X

j=1

uk;j+ z

k X

j=1

vk;j);

jp,qj=j

X

k=1

(wk,zk)j:

(3.10)

These relationship show that the parameters p;q will aect the eigenstructure of C.

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is unitarily similar to a real diagonal matrix. Another direct consequence is that for a real eigenvalue the largest size of the associated Jordan block is not larger than 2minfp;qg+ 1, and for a nonreal eigenvalue the largest size of the associated Jordan

block is not larger than minfp;qg. Furthermore, it is clear that ifjp,qj6= 0, then C

must have real eigenvalues with at leastjp,qjodd sized Jordan blocks.

The real structured Jordan canonical form for a real p;q-symmetric matrix, under

real p;q-orthogonal transformations can be obtained analogously. Theorem 3.6. Let C be a real

p;q-symmetric matrix with pairwise dierent

real eigenvalues 1

;:::;

1 ;:::;

with positive

imaginary parts. Then there exists a real p;q-orthogonal matrix

U, such that

i) The blocks with index c, associated with nonreal eigenvalues, are R

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These havefor k= 1;:::; the substructures

even sized Jordan blocks of sizes 2p k ;1 sizedJordanblocks ofsizes2l

k ;1+ 1 ;:::;2l

k ;xk+ 1 . Foreach real eigenvalue

k

there are a)t

k

even sizedJordanblocksof sizes2q k ;1

;:::;2q k ;t

k

corresponding tothe structure inertia indices

k ;1

odd sized Jordanblocks of sizes2u k ;1+ 1

;:::;2u k ;w

k+ 1

corresponding tothe structureinertia index 1;

c) z k

odd sized Jordan blocks of sizes 2v k ;1+ 1

;:::;2v k ;z

k+ 1

corresponding to the structureinertia index ,1.

Proof. For real eigenvalues 1

;:::;

, using I.b, ii) of Proposition 2.3, as in the

proof of Theorem 3.3, there exists a real matrixU r:= [

which is the real version of (3.9).

For a nonreal eigenvalue of C with a Jordan block N r(

), by Proposition 2.3

there is a real matrix ^U such that

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and ifris odd then set

U := ^Udiag(Ir+1;P^ r ,1

2

1;1) 2 6 6 4

p 2 2 Ir

,1 0 ,

p 2 2 Ir

,1 0

0 1 0 0

0 0 0 ,1

p 2 2 Ir

,1 0 p

2 2 Ir

,1 0

3 7 7 5:

By Proposition 2.1 we have that ( ^Ps

1;1) ,1= ^P

s

1;1= ( ^Ps

1;1)

T_; _{( ^}_P_s 1;1)

T₍_N_s_{())( ^}_P_s

1;1) = (Ns())

T_;

and some simple calculations yield

UT_p;q_U ₌

Ir 0

0 ,Ir

; CU =U

A T

,TT B

;

where, ifr= 2s, then

A=N+

s ((Re)I2) +Er; B=N +

s((Re)I2)

,Er; T =,N ,

s ((Im)J1) +Er;

withEr=1 2

0 0 0 1;1

; and ifr= 2s+ 1, then withJ1=

0 1

,1 0

,

A=

"

N+

s ((Re)I2) p

2 2 er

,2 p

2 2 eTr

,2 Re

#

; B=

"

N+

s((Re)I2) p

2 2 er

,1 p

2 2 eTr

,1 Re

#

;

T =

" ,N

,

s ((Im)J1) ,

p 2 2 er

,1 p

2 2 eTr

,2

,Im #

:

Now as for the case of real eigenvalues in the proof for Theorem 3.3, for nonreal eigenvalues1;:::; there exists a real matrix

Uc:= [Vc;Wc] such that UTcp;qUc= m;m; CUc=Uc

R+

c Tc

,TTc R ,

c

;

whereR+

c,Tc,R,

c are in the asserted forms andm=P

k=1( Ps

k

j=12pkj+ Px

k

j=1(2lkj+

1)).

SetU = [Vc;Vr; Wc;Wr] then analogously we get that U is real p;q-orthogonal

andU ,1

CU has the form (3.11).

In this section we have obtained real and complex structured Jordan canonical forms for p;q-Hermitian matrices. In the next section we present analogous results

for p;q-skew Hermitian matrices.

4.

p;q

-skew Hermitian matrices.

In this section we discuss structured

Jor-dan canonical forms for p;q-skew Hermitian matrices. The construction is similar to

that for p;q-Hermitian matrices discussed in Section 3 and therefore we omit much

of the detail. The essential dierence is that the role of the real eigenvalues is now taken by the purely imaginary eigenvalues.

Analogous to the p;q-Hermitian matrices by employing the results in

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Theorem 4.1. Let C be a

p;q-skew Hermitian matrix with pairwise dierent

purely imaginary eigenvalues 1

;:::;

1 ;:::;

with positive real parts. Then there exists a nonsingular matrix U such that U

,1

CU = diag(R + c

;R , c

;R g)

;

i) The diagonal blocks with indexc, associated with eigenvalues not on the imaginary

axis, are

R +

c = diag( H

1(

1) ;:::;H

(

)) ; R

,

c = diag( H

1( ,

1) ;:::;H

( ,

)) ;

where for k= 1;:::;we have H

k(

k) =

k I+H

k ; H

k( ,

k) = ,

k I +H

k ; H

k = diag( N

p k ;1

;:::;N p

k ;s k)

:

ii) The block R

g, associated with purely imaginary eigenvalues, has the form R

g= diag( M

1(

1) ;:::;M

(

)) ;

whereM k(

k) =

k

I+M

k and for

k= 1;:::; we haveM

k= diag( N

qk ;1, :::,N

qk ;t k).

The matrixU has the form

U H

p;q U =

2 4

0 W

c 0

W H

c 0 0

0 0 W

g 3 5 ;

where

W

c = diag( P

H 1

;:::;P H

)

; W

g= diag( W

g 1

;:::;W g )

;

and for k = 1;:::; we have P H

k = diag( P

p k ;1

;:::;P p

k ;s

k), and with Ind(

k) = f

k ;1 ;:::;

k ;tk

gfor k= 1;:::; we haveW g

k = diag(

k ;1 P

qk ;1 ;:::;

k ;tk P

qk ;t k). Theorem 4.2. Let C be a real

p;q-skew symmetric matrix with pairwise

dif-ferent nonzero purely imaginary eigenvalues 1

;:::;

with positive imaginary parts,

pairwise dierent eigenvalues 1

;:::;

with positive real and imaginary parts and

pairwise dierent real positive eigenvalues 1

;:::;

. (Note that when the spectrum

contains

k, it also contains ,

k, if it contains

j then also ,

j and if it contains

j then also ,

j ;

j ;,

j. Furthermore0 may be an eigenvalue.) Then there exists a

real nonsingular matrixU such that U

,1

CU = diag(R + c

;R , c

;R g)

:

i) The blocks with indexc, associated with eigenvalues not on the imaginary axis, are R

+

c = diag( ^ R + c

;R~ +

c), with ^ R +

c = diag( K

1(

1) ;:::;K

(

)) where for

k = 1;:::;

we have K k(

k) =

k

I+K k and

K

k = diag( N

fk ;1 ;:::;N

fk ;l

k). Analogously ~ R + c =

diag(H 1(1)

;:::;H

()), where for

k= 1;:::;we have k=

Re

k Im

k ,Im

k Re

k

andH

k(k) = diag( N

p k ;1(

k) ;:::;N

p k ;s

k( k)).

The block R ,

c = diag( ^ R , c

;R~ ,

c ), has the same substructure as R

+

c just replacing

j with ,

j andj by ,

j.

ii) The block R

g, associated with purely imaginary eigenvalues, has the structure R

g= diag( M

1((Im

1) J

1) ;:::;M

((Im

) J

1) ;M

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where for k= 1;:::; we have M

k((Im

k) J

1) = diag( N

qk ;1((Im

k) J

1) ;:::;N

qk ;t k((Im

k)

J 1))

;

and where

M

0= diag( N

2g 1

+1 ;:::;N

2g a

+1 ;N

2h 1

;N 2h

1 ;:::;N

2h b

;N 2h

b)

is the structure associated with the eigenvalue0. The matrixU has the form

U T

p;q U =

2 4

0 W

c 0

W T

c 0 0

0 0 W

g 3 5 ;

whereW

c = diag( ^ W c

;W~ c) with

^

W

c= diag( P

K 1

;:::;P K

) ; W~

c = diag( P

H 1

1;1 ;:::;P

H

1;1) ;

and where

P K

k = diag( P

f k ;1

;:::;P f

k ;l k)

; P

H

k= diag( P

p k ;1

;:::;P p

k ;s k)

:

The block W

g has the form W

g = diag( W

g 1

;:::;W g

;W

0), where for

k= 1;:::; and

Ind( k) =

f k ;1

;:::; k ;t

k

gwe have W g

k = diag( P

q k ;1

k ;1 ;:::;P

q k ;t

k

k ;t k), with

k ;j =

k ;j I

2 if q

k ;j is odd and k ;j = (Im

k ;j) J

1 if q

k ;j is even.

Finally forInd(0) =f 0 1

;:::; 0 a

;i;,i;:::;i;,igwe have W

0= diag(

0 1 P

2g1+1 ;:::;

0 a P

2ga+1 ;

0 P 2h1 P

T

2h1 0

;:::;

0 P 2hb P

T

2hb 0

):

After determining the Jordan structure under non p;q-unitary similarity

trans-formations we now derive the corresponding structured canonical form under p;q

-unitary transformations.

Theorem 4.3. Let C be a

p;q-skew Hermitian matrix with pairwise distinct

eigenvalues 1

;:::;

with positive real parts and pairwise distinct

1 ;:::;

with

real part zero. Then there exists ap;q-unitary matrix

U, such that

U ,1

CU = 2 6 6 4

R c

T c R

+ g

T g T

H c

R c T

H g

R , g

3 7 7 5 :

(4.1)

For the dierent blocks we have the following substructures.

i) The blocks with indexc, associated with eigenvalues not on the imaginary axis, are R

c = diag( R

c 1

;:::;R c ) and

T

c = diag( T

c 1

;:::;T c

) where for

k= 1;:::; R

c

k = diag( N

, p

k ;1( iIm

k) ;:::;N

, p

k ;s k(

iIm k))

; T

c k =

,diag(N + pk ;1(Re

k)

;:::;N + pk ;s

k

(Re k))

:

ii) The blocks with indexg, associated with purely imaginary eigenvalues, are R

+

g = diag( C

1 ;:::;C

) ; R

,

g = diag( D

1 ;:::;D

) ; T

g= diag( F

1 ;:::;F

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where for k= 1;:::; the substructures are

Ck = diag(Cek;C+

k;C,

k); Dk= diag(Dek;D+

k;D,

k); Fk = diag(Fek;F+

k ;F,

k );

with further partitioning

C_ek= diag(N,

qk ;1(k) + 12ik; 1eq

k ;1eHq

k ;1;:::;N ,

qk ;t k

(k) + 12ik;tkeqk ;t keHq

k ;t k

); D_ek= diag(N,

qk ;1(k) ,

1 2ik;1eq

k ;1eHq

k ;1;:::;N ,

qk ;t k

(k),

1

2ik;tkeqk ;t keHq

k ;t k

); F_ek= diag(,N

+

qk ;1+ 12ik; 1eq

k ;1eHq k ;1;:::;

,N +

qk ;t k+ 12

ik;tkeqk ;t keHq

k ;t k

); C+

k = diag

"

N,

uk ;1(k) p

2 2 eu

k ;1 ,

p 2 2 eHu

k ;1 k #

;:::;

"

N,

uk ;w k

(k) p 2 2 eu

k ;w k ,

p 2 2 eHu

k ;w

k k

#!

; D+

k = diag(N,

uk ;1(k);:::;N ,

uk ;w k(

k));

F+

k =,diag

"

N+

uk ;1 p

2 2 eHuk ;1

#

;:::;

"

N+

uk ;w k p

2 2 eHuk ;w

k # !

; C,

k = diag(N,

vk ;1(k);:::;N ,

vk ;z k

(k));

D,

k = diag

"

k p 2 2 eHv

k ;1 ,

p 2 2 ev

k ;1 N ,

vk ;1(k) #

;:::;

"

k p 2 2 eHvk ;z

k ,

p 2 2 ev

k ;z

k N

,

vk ;z k

(k)

# !

;

F,

k = diag([

p

2 2 evk ;1;

,N +

vk ;1];:::[ p

2 2 evk ;z

k; ,N

+

vk ;z k]):

Eachk(,k) hasskJordan blocks of sizespk;

1;:::;pk;s

k. Each purely imaginary

eigenvaluek has

a)tk even sized Jordan blocks of sizes2qk;1;:::;2qk;t

k with the corresponding structure

inertia indicesi(,1)q k ;1+1

k;1;:::;i( ,1)q

k ;t k

+1

k;tk;

b) wk odd sized Jordan blocks of sizes2uk;1+ 1;:::;2uk;w

k+ 1 corresponding to the

structure inertia indices(,1)u

k ;1+1;:::;( ,1)u

k ;w k

+1;

c) zk odd sized Jordan blocks of sizes 2vk;1+ 1;:::;2vk;z

structure indices (,1)v

k ;1;:::;( ,1)v

k ;z k.

Proof. For all eigenvalues1;:::;, by Theorem 4.1 there is a matrix ^ Uc such

that

^

UHcp;qU^=

0 Wc

W_cH ₀

; CU^c= ^Uc

R+

c 0

0 R,

c

;

whereWc,R+

c,R,

c are dened in Theorem 4.1. LetUc:= ^Ucdiag(Im;W ,1

c )m, where

m is dened in (2.1) andm =P

k=1 Ps

k

j=1pkj. By using i), ii) of Proposition 2.2

and (2.2), (2.3) we have

UHcp;qUc= m;m; CUc=Uc

Rc Tc

T_cH _R_c

;

(4.2)

whereRc,Tc are as asserted.

Now we consider the purely imaginary eigenvalues. As for the real eigenvalues of p;q-Hermitian matrices we rst focus on one Jordan blockNr() with purely

imaginary. According to Proposition 2.3 for this block there is a matrix ^U such that ^

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where =1 if r is odd and =i if r is even. Similarly if r is even let U :=

Applying these formulas to all purely imaginary eigenvalues 1

;:::;

,

analo-gous to the real eigenvalue case in Theorem 3.3 for p;q-Hermitian matrices we can

construct a matrixU

g such that U

where R +

g are dened in the theorem and n

The p;q-unitary matrix

U can then be generated fromU c,

U

g, and by combining

above relation with (4.2) we have (4.1).

As the nal result in this section we present the real version of Theorem 4.3.

Theorem 4.4. Let C be a real

p;q-skew symmetric matrix with pairwise

dis-tinct real positive eigenvalues 1

;:::;

, pairwise distinct eigenvalues

1 ;:::;

with

positive real and imaginary parts and pairwise distinct purely imaginary eigenvalues

1

;:::;

with positive imaginary parts. (Note that we then also have eigenvalues ,

0 may be another eigenvalue.)

Then there exists a real p;q-orthogonal matrix

U, such that

where the dierent blocks have the following substructures:

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ELA

and for k= 1;:::;,

ii) The blocks with indexg, associated with the purely imaginary eigenvalues, are R

with the partitioning

C

Finally, the blocks with index 0, associated to the eigenvalue 0, are

C

with substructures

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C+ 0 = diag

"

N,

g1 p

2 2 eg

1 ,

p 2

2 eTg1 0 #

;:::;

"

N,

ga p

2 2 eg

a ,

p 2

2 eTga 0 #!

;

D+

0 = diag(N ,

g1;:::;N ,

ga); F + 0 =

,diag

N+

g1 p

2 2 eTg1

;:::;

N+

ga p

2 2 eTga

;

C,

0 = diag(N ,

h1;:::;N ,

hb); F ,

0 = diag [ p

2 2 eh1;

,N +

h1];:::;[ p

2 2 ehb;

,N +

hb] !

;

D, 0 = diag

"

0 p

2 2 eTh

1 ,

p 2 2 eh

1 N

,

h1 #

;:::;

"

0 p

2 2 eTh

b ,

p 2 2 eh

b N

,

hb #!

:

Each nonzero real eigenvaluek (,k) haslk Jordan blocks of sizesfk;

1;:::;fk;l k

and each nonreal eigenvalue k (,k;k;,k) that is not on the imaginary axis has

sk Jordan blocks with sizespk;1;:::;pk;s k.

For each nonzero purely imaginary eigenvalue k (,k) we have

a) tk even sized Jordan blocks of sizes 2qk;1;:::;2qk;t

k with the corresponding

struc-ture inertia indicesi(,1)q k ;1+1

k;1;:::;i( ,1)q

k ;t k

+1

k;tkfor k andi( ,1)q

k ;1

k;1;:::,

i(,1)q k ;t

kk;t k for

,k;

b) wk odd sized Jordan blocks of sizes2uk;1+ 1;:::;2uk;w

structure inertia indices(,1)u

k ;1+1;:::;( ,1)u

k ;w k

+1;

c) zk odd sized Jordan blocks of sizes 2vk;1+ 1;:::;2vk;z

structure indices (,1)v

k ;1;:::;( ,1)v

k ;z k.

The zero eigenvalue has 2c even sized Jordan blocks with sizes of 2x1;2x1, :::,

2xc;2xcwith corresponding structure inertia indicesi;,i;:::;i;,i, anda+bodd sized

Jordan blocks, where a of them have sizes 2g1+ 1;:::;2ga+ 1 with the

correspond-ing structure inertia indices (,1)g 1

+1;:::;( ,1)g

a

+1 and b of them have sizes 2h 1+

1;:::;2hb+ 1 with the corresponding structure inertia indices (,1)h 1;:::;(

,1)h b.

Proof. As in the previous proofs we need to study the canonical forms of the non purely imaginary and purely imaginary eigenvalues separately. Here for the latter case we have to deal with two subcases, the nonzero and zero eigenvalues. For non purely imaginary eigenvalues by Theorem 4.2 there is a real matrix ^Uc such that

^

UTcp;qU^c=

0 Wc

WTc 0

; CU^c = ^Uc

R+

c 0

0 R,

c

:

LetUc:= ^Ucdiag(Im;W ,1

c )m, wherem:=P

k=1 Pt

k

j=1fk;j+ P

k=1 Ps

k

j=12pk;j. By

using Proposition 2.2 we can verify that

UTcp;qUc= m;m; CUc=Uc

Rc Tc

T_Tc Rc

;

whereRc,Tc are as asserted.

For nonzero purely imaginary eigenvalues1;::: we rst consider a single

Jor-dan blockNr(ImJ1). By Proposition 2.3 there exists a real matrix ^U such that

^

UT_p;q_U_^₌_P_r; CU^= ^UNr(ImJ 1);

where =I2 ifr is odd and = (Im)J1 isr even, and is the structure inertia

index corresponding toNr(). If ris even, then we setU := ^Udiag(Ir;((Im)Pr 2

J1) ,1)

r

2 and obtain

UT_p;q_U ₌

Ir 0

0 ,Ir

; CU =U

N,

s ((Im)J1) +Er

,N +

s (02) +Er ,N

+

s(02)

,Er N

,

s ((Im)J1) ,Er

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ELA

whereE r=

1 2

0 0 0 J

1

and = (,1) r 2

i. Ifris odd, then we set U := ^Udiag(I

r+1

;(Pr ,1 2

I 2)

,1)(^ r ,1

2 I

2)

we have

U T

p;q U =

2 4

I r,1

I 2

,I r,1

3 5 ;

CU =U 2 6 6 6 6 6 4

N , r ,1

2

((Im)J 1)

0

p 2 2

I 2

,N + r ,1

2

(02)

0 , p

2 2

I

2 (Im

)J

1 0

, p

2 2

I 2 ,N

+ r ,1

2

(02)

0

, p

2 2

I 2

N , r ,1

2

((Im)J 1)

3 7 7 7 7 7 5 ;

where = (,1) s+1

2

. Based on these properties we can construct a real matrixU g

such that

U T g

p;q U

g= n 1

;n 2

; CU

g= U

g

^

R +

g ^

T g

^

T T

g ^

R , g

;

where ^

R +

g = diag( C

1 ;:::;C

) ; R^

,

g = diag( D

1 ;:::;D

) ; T^

g= diag( F

1 ;:::;F

) ;

andC k

;D k

;F k (

k= 1;:::;) are in the asserted forms and n

1= 2 X k =1

(w k+

tk X j=1 q

k ;j+ wk X j=1 u

k ;j+ zk X j=1 v

k ;j) ;

n 2= 2

X k =1

(z k+

tk X j=1 q

k ;j+ wk X j=1 u

k ;j+ zk X j=1 v

k ;j) :

For the eigenvalue zero we have distinguished between even and odd sized Jordan blocks. For odd sized Jordan blocks asN

rthere exists a real matrix ^

U such that

^

U T

p;q^

U =P

r

; CU^= ^UN r

:

As in the purely imaginary case in the proof of Theorem 4.3 we then can generate a real matrixU from ^U such that

U T

p;q U =

2 4

Ir ,1

2 0 0

0 0

0 0 ,Ir ,1 2

3 5

;CU =U 2 6 6 4

N , r ,1

2

p 2 2

er ,1 2

,N + r ,1

2 ,

p 2 2

e T

r ,1 2

0 ,

p 2 2

e T r ,1

2 ,N

+ r ,1

2 ,

p 2 2

er ,1 2

N , r ,1

2 3 7 7 5 ;

with= (,1) r +1

2

. By Proposition 2.3, even sized Jordan blocksN

rmust appear in

pairs. More precisely for each pair ofN r

;N

rthere is real matrix ^

U such that

^

U T

p;q^ U =

0 P r P

T

r 0

; CU^= ^U

N

r 0

0 N r

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Hence withU := ^Udiag(I r

;P ,1

r )

r we have U

T p;q

U =

I

r 0

0 ,I r

; CU =U

N , r

,N + r ,N

+ r

N , r

:

Based on these facts for the eigenvalue zero there exists a real matrixU

z such that U

T z

p;q U

z= n

0 1 ;n

0 2

; CU

z= U

z

C 0

F 0 F

T 0

D 0

;

where C 0,

F 0,

D

0 are in the asserted forms, and n

0 1 =

a+ P

c k =12

x k +

P a k =1

g k + P

b k =1

h k,

n 0 2=

b+ P

c k =12

x k+

P a k =1

g k+

P b k =1

h k.

Finally by combining all these cases we can generate a real p;q-orthogonal matrix U fromU

c, U

g, U

zwhich satises (4.3).

We have seen that the results for p;q-Hermitian and p;q-skew Hermitian

ma-trices are quite similar, which was to be expected, since both classes have an algebra structure. In the next section we now study the canonical forms for matrices in the associated Lie group of p;q-unitary matrices.

5.

p;q

-unitary matrices.

In the previous two sections we have studied

struc-tured Jordan canonical forms for p;q-Hermitian and p;q-skew Hermitian matrices.

Each class has an algebra structure, the p;q-Hermitian matrices form a Jordan

al-gebra and the p;q-skew Hermitian matrices a Lie algebra. The Lie group associated

with these two algebras is the class of p;q-unitary matrices. In order to derive

struc-tured canonical forms for this group analogous to the results for the algebras, we can make use of the Cayley transformation.

Lemma 5.1. If Ais

p;q-unitary and1

62(A) then the Cayley transformation

of B

B=(A) = (A+I)(A,I) ,1

(5.1)

isp;q-skew Hermitian. Conversely, if

Ais

p;q-skew Hermitian then

B as in (5.1)

isp;q-unitary.

Proof. We only prove the result for the case that A is

p;q-unitary. The other

direction follows form the fact that((A)) =A.

SinceAis

p;q-unitary, p;q A=A

,H

p;q. By this relation

p;q B=

p;q(

A+I)(A,I) ,1= (

A ,H+

I) p;q(

A,I) ,1

= (A ,H+

I)(A ,H

,I) ,1

p;q= ( I+A

H)( A

,H)( A

H)( I,A

H),1 p;q

= (A+I) H(

I,A)

,H

p;q= ,B

H p;q=

,( p;q

B) H

:

ThereforeBis

p;q-skew Hermitian.

Using the Cayley transformationthe canonical forms of

p;q-unitary matrices

(if 1 is not an eigenvalue) can be easily obtained from the canonical form of the corresponding p;q-skew Hermitian matrix discussed in Section 4. However, if we

Cayley transform the canonical form it is usually not a canonical form again and we need further reductions to obtain again the canonical form. But, obviously it suces to further reduce each Jordan block separately. Before discussing these reductions, we rst split the Jordan structure of a p;q-unitary matrix

Ginto two parts, the part

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ELA

Lemma 5.2. LetG be a

p;q-unitary matrix that has1 as an eigenvalue. Then,

there exists a nonsingular matrix Y, such that Y

H

Y = diag( p1;q1

; p2;q2)

; Y

,1

GY = diag(G 1

;G 2)

;

wherep 1+

p 2=

p;q 1+

q 2=

q,G

1 is p1;q1-unitary with 1 62(G

1) and G

2 isp2;q2

-unitary and has1 as only eigenvalue.

Furthermore, ifG is real, thenY can be chosen real, so that G 1

;G

2 are also real.

Proof. Let ^Y be a nonsingular matrix such that GY^= ^Ydiag( ^G

1 ;G^

2) =: ^ YG;^

with 1 62 ( ^G

1) and ( ^ G

2) =

f1g. Then we have ^Y H

G H = ^

G H^

Y

H and, using the

p;q-unitarity of

G we have the discrete Lyapunov (or Stein) equation

^

G H( ^

Y H

p;q^ Y) ^G= ^Y

H p;q^

Y:

(5.2)

By the diagonal block form of ^G and the eigenvalue splitting, the solution of (5.2) has

also block diagonal form, i.e., ^Y H

p;q^

Y= diag(T 1

;T

2). Note that ^ Y

H p;q^

Yas well as T

1 ;T

2 are nonsingular Hermitian. Therefore, there exist nonsingular matrices Z

1 ;Z

2

such that

Z H 1

T 1

Z 1= p

1 ;q

1

; Z

H 2

T 2

Z 2= p

2 ;q

2 :

To nish the proof, we setY = ^Ydiag(Z 1

;Z 2),

G 1=

Z ,1 1 ^

G 1

Z 1and

G 2=

Z ,1 2 ^

G 2

Z 2.

The real case is clear, since 1 is a real eigenvalue.

It is well known, that Cayley transformation directly leads to a rational relation-ship between the eigenvalues, i.e., if 6= 1 is an eigenvalue of a

p;q-unitary matrix G, then=() =

+1

,1 is an eigenvalue of the Cayley transformation

(G) and we

have the following well-known facts.

Proposition 5.3. Let G be

p;q-unitary with 1

62(G). Set C =(G) and let 2(G) and=()2(C). Then

i)6= 1;,1.

ii) andhave the same algebraic and geometric multiplicities.

iii)jj= 1 if and only ifis purely imaginary.

iv) If2(C) is not purely imaginary, then,=(

,1) and, furthermore,

;,2

(C) if and only if ; ,1

2 (G). In order to further reduce Cayley transformed

Jordan blocks we need the following result.

Lemma 5.4. Let N r(

) be a Jordan block with 6= 1 and let = (). Then

there exists a nonsingular upper triangular matrix X

r such that X

,1 r

(N r(

))X r=

N r(

);

ande T r

X r

e r

6

= 0. Proof. See, e.g., [8].

We are now prepared to present block by block the transformations of the results in Section 4.

p;q-unitary matrix and let

N() =I+N with N =

diag(N r

1 ;:::;N

r

s) be the Jordan structure of

G corresponding to2(G) withjj6=

1. Then there exists a full rank matrixU such that

U H

p;q U =

0 ^P N

^

P H

N 0

; GU =U

N() 0

0 N() ,1

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ELA

and ,1

2(G) has the same algebraic and geometric multiplicities as.

If G is real then we have two cases:

i) If is real then there exists a real full rank matrixU such that

U T

p;q U =

0 ^P N

^

P T

N 0

; GU =U

N() 0

0 (N()) ,1

:

ii) If is nonreal then there exists a real full rank matrixU such that

U T

p;q U =

0 P^ N

1;1

^

P T N

1;1 0

; GU =U

N(,) 0

0 (N(,)) ,1

;

with, =

Re Im ,Im Re

:

Proof. We may assume without loss of generality that 162(G). Otherwise by

Lemma 5.2 we can consider the smaller size matrix G 1. If

is the Cayley

trans-formation, then by Lemma 5.1, C = (G) is

p;q-skew Hermitian. Furthermore, = ()2(C) and by Proposition 5.3 ii), iv), is not purely imaginary and the

associated Jordan structure associated with is I+N. Applying Proposition 2.3

there exists a matrix ^U such that

^

U H

p;q^ U =

0 P N P

H

N 0

; CU^ = ^U

N() 0

0 N(,)

:

With ~U = ^Udiag(I;P ,1

N ) and, since P

N NP

H N =

,N

H, we have

~

U H

p;q~ U =

0 I

I 0

;CU~= ~U

N() 0

0 ,(N()) H

:

Using the Cayley transformation then we have

GU~= ~U =

(N()) 0

0 (,(N()) H)

:

Note that

(,N()) H) = (

,N() H+

I)(,N() H

,I) ,1

=f(N(),I)(N() +I) ,1)

g H

=f(N())g ,H

:

Applying Lemma 5.4, there exists a nonsingular matrixX = diag(X r1

;:::;X

rs) such

that X ,1

(N())X =N(). ObviouslyX H

f(N())g ,H

X ,H =

N() ,H

: Setting V = ~Udiag(X;X

,H) we obtain V

H p;q

V =

0 I

I 0

; GV =V

N() 0

0 N() ,H

;

and takingU =Vdiag(I;P^

N) nishes the proof in the complex case..

Since the Cayley transformation of a real matrix is also real, we can apply Propo-sition 2.3 to get the result for the real case.

This result shows that for the eigenvalues of a p;q-unitary matrix that are not

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ELA

matrix, only half of these eigenvalues have the classical Jordan structure, while for the other half of the eigenvalues we have to involve the inverses of Jordan blocks.

For eigenvalues withjj= 1 the canonical structure is even more complicated. If

we restrict the chains of root vectors to have the proper structures coming form a p;q

-skew Hermitian matrices as in Proposition 2.3 then no Jordan block will appear in the canonical form. We can do further reductions for which we will need the following simple result.

Lemma 5.6. Given a vector t = [t1;:::;t r]

T and t

r

6

= 0 then there exists a nonsingular upper triangular Toeplitz matrixT such that T

,1 t=e

r.

Proof. See [8].

We now study the reduction of Cayley transformed blocks arising form unimodular eigenvalues.

p;q-unitary matrix and let

2(G) with jj = 1 and 6= 1. LetN

r(

) be a single Jordan block, then there exists a full rank matrixU such

that

U H

p;q U = ^P

r

; GU =U

N s(

) ie

s e

H

1N s(

) ,1

0 N

s( )

,1

;

(5.3)

if r= 2sand U

H p;q

U =P^ r

;

(5.4)

GU =U 2 4

N s(

) e

s

1, e

s e

H

1N s(

) ,1

0 ,e

H

1N s(

) ,1

0 0 N

s( )

,1 3 5 ;

if r= 2s+ 1.

Here = (,1) s

i with 2 fig if r = 2s and = (,1) s+1

, 2 f1g if r = 2s+ 1 where is the structure inertia index of the corresponding eigenvalue =().

If G is real then we have two cases:

i) If 6=,1, then with , =

Re Im ,Im Re

, ^P2=

0 1 1 0

and

S() =,

1 2

"

1 Im

1,Re

Im

1,Re ,1

# ;

there exists a real full rank matrix U such that if r= 2s, then

U T

p;q U = ^P

r

1

;1

; GU =U 2 4

N s(,)

,

0 0 ^

P2 0

N s(,)

,1

0 N

s(,) ,1

3 5 ;

(5.5)

and if r= 2s+ 1, then

U T

p;q U =

2 4

0 0 ^P s

1

;1

0 I2 0

^

P s

1

;1 0 0

3 5 ;

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ELA

GU =U 2 6 6 4

N

s(,) 0,

0 0

S() 0

N s(,)

,1

0 , [,

1;1 ;0]N

s(,) ,1

0 0 N

s(,) ,1

3 7 7 5 :

ii) If=,1, then there exists a real full rank matrix U such that

U T

p;q U =

0 ^P r

^

P T

r 0

; GU =U

N r(

,1) 0

0 N

r( ,1)

,1

;

(5.7)

if ris even and

U T

p;q

U =P^ r

; GU =U 2 4

N s(

,1) ,e s

, 1 2 e

s e

T 1

N s(

,1) ,1

0 ,1 ,e T 1

N s(

,1) ,1

0 0 N

s( ,1)

,1 3 5 ;

(5.8)

if r = 2s+ 1. Here = (,1) s+1

and is the structure inertia index of 0

corre-sponding to(G).

Proof. We may again assume without loss of generality that 162 (G) and set C =(G). By Proposition 5.3 the corresponding=() now is purely imaginary,

andC has the Jordan blockI+N

r. Applying Proposition 2.3 there exists a matrix

^

U such that

^

U H

p;q^

U =P

r

; CU^= ^UN r(

):

(5.9)

Ifr= 2s then2figand we partition

^

U H

p;q^ U =

0 P

s

(P s)

H 0

; N

r( ) =

N

s( ) e

s e

H 1

0 N

s( )

:

Applying the Cayley transformation we obtain

GU^ = ^U(N r(

)):

Using the notation ^N s(

) =(N s(

)) and the property that (N s(

),I) ,1=

1 2( ^

N s(

), I) we obtain

(N r(

)) =

^

N s(

) 1 2(

I,N^ s(

))e s

e H 1( ^

N s(

),I)

0 N^

s( )

:

Setting ~U = ^Udiag(I s

;(P s)

,1), then

~

U H~

U =

0 I

I 0

; GU~ = ~U

^

N s(

) i

2( I,N^

s( ))e

s e

H s ^

N s(

) ,H(

I,N^ s(

)) H

0 N^

s( )

,H

:

By Lemma 5.2, there exists a nonsingular upper triangular matrix X such that X

,1^ N s(

)X = N s(

). Since the last component of t := X ,1(

p 2 2

e

s) is nonzero,

by Lemma 5.5 there exists a nonsingular upper triangular Toeplitz matrix T such

that T ,1

t=e

s. Setting

Y =X(I,N s(

))T and U = ~Udiag(Y;Y ,H^

P

s), we obtain

(5.3), since (I,N s(

))T commutes with N s(

) and ^P ,1 s

N s(

) ,H^

P s=

N s(

) ,1.

Ifris odd, following (5.9) we partition

^

U H

p;q^ U =

2 4

0 0 P

s

0 0

(P s)

H 0 0

3 5

; N

r( ) =

2 4

N s(

) e

s 0

0 e

H 1

0 0 N

s( )

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ELA

and as before we obtain

GU^= ^U 2 4

^

N s(

) ,1

2 ( I,N^

s( ))e

s 1,

4 ( I,N^

s( ))e

s e

H 1( ^

N s(

),I)

0

1, 2

e H 1( ^

N s(

),I)

0 0 N^

s( )

3 5 :

With ~U = ^Udiag(I s+1

;(P s)

,1) we then have

~

U H

p;q~ U =

2 4

0 0 I s

0 0 I

s 0 0

3 5 ;

CU~ = ~U 2 4

N s(

) ,1

2 ( I,N^

s( ))e

s ,1

4

(I,N^ s(

))e s

e H s ^

N s(

) ,H(

I,N^ s(

) H)

0

,1 2

e H s ^

N s(

) ,H(

I ,N^ s(

) H)

0 0 N^

s( )

,H

3 5 :

SettingY = 1,

2

X(I,N s(

))T andU = ~Udiag(Y;1;Y ,H

P^

s), we have (5.4).

IfG is real, then the real forms (5.5), (5.6) and (5.7), (5.8) can be derived in the

similar way. Note that if =,1 then the corresponding eigenvalue ofC=(G) is 0.

So far we have restricted ourselves to the Jordan structure associated with eigen-values not equal to 1. For the eigenvalue 1 we give a separate analysis.

Lemma 5.8. Let G be a

p;q-unitary matrix and let N

r(1) be a Jordan block of G. Then there exists a full rank matrixU such that

U H

p;q U= ^P

r

; GU =U

N s(1)

,ie s

e H 1

N s(1)

,1

0 N

s(1) ,1

;

if r= 2sand if r= 2s+ 1, then

U H

p;q

U =P^ r

; GU =U 2 4

N s(1)

e s

, 1 2 e

s e

H 1

N s(1)

,1

0 1 ,e

H 1

N s(1)

,1

0 0 N

s(1) ,1

3 5 :

(5.10)

Here = (,1) s

i with 2 fig if r = 2s and = (,1) s+1

with 2 f1g if r= 2s+ 1.

If G is real, then there exists a real matrixU such that

U T

p;q U = ^P

2r

; GU =U

N

r(1) 0

0 N

r(1) ,1

;

if ris even and if r= 2s+ 1 we have again (5.10).

Proof. By Lemma 5.2 we may assume without loss of generality that (G) = f1g. Otherwise we work on the small size matrix G

2. We cannot use the Cayley

transformationbut a dierent rational transformation ^(z) = (1,z)(1+z) ,1. If

A

is p;q-unitary then

B= ^(A) is

p;q-skew Hermitian and conversely. With this new

transformation we obtain the proof analogous to the proof of Lemma 5.7.

Using these results, analogous to the case for the Jordan and Lie algebras we can show the following structured canonical forms for both complex and real p;q-unitary

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Theorem 5.9. LetG be a

p;q-unitary matrix

G, let 1

;:::;

be the pairwise

dierent eigenvalues of modulus less than one and let 1

;:::;

be the pairwise

dif-ferent eigenvalues of modulus one. Then there exists a nonsingular matrix U such

that

i) The diagonal blocks R c

;R i

c, associated with eigenvalues not on the unit circle, are R

ii) The diagonal block R

u associated with the unimodular eigenvalues are R

=1and furthermore e

The matrixU has the form

U

Each eigenvalue k (

k

,1

) has s

k Jordan blocks of sizes p

k ;1 ;:::;p

k ;s

k and each

unimodular eigenvalue k has

a)t

k even sized Jordan blocks of sizes 2q

k ;1 ;:::;2q

k ;t

k corresponding to the structure

inertia indices(,1) q

k odd sized Jordan blocks of sizes 2r

k ;1

+1;:::;2r k ;wk

+1corresponding to the

structure inertia indices(,1) r

p;q-orthogonal matrix, let

1 ;:::;

be pairwise

dierent real eigenvalues of modulus less than one, let 1

;:::;