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

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 THAT ARE HERMITIAN, SKEW HERMITIAN OR
UNITARY WITH RESPECT TO INDEFINITE INNER PRODUCTS
VOLKER MEHRMANNy AND HONGGUO XUy

Abstract. For inner products dened by a symmetric indenite matrix  , canonical forms for
real or complex  -Hermitian matrices,  -skew Hermitian matrices and  -unitary matrices
are studied under equivalence transformations which keep the class invariant.
p;q

p;q

p;q

p;q

Key words. structured eigenvalue problems, Lie group, Lie algebra, Jordan algebra
AMS subject classications. 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 transformation 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 reduction 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 numerical 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 transformations. 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 classied 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 indenite 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 dened via
 Received by the editors on 13 October 1998. Accepted for publication on 30 April 1999. Handling
Editor: Daniel Hershkowitz.
y Fakult
at fur Mathematik, TU Chemnitz, D-09107 Chemnitz, FR Germany. This work has been
supported by Deutsche Forschungsgemeinschaft within SFB393 `Numerische Simulation auf massiv
parallelen Rechnern'
67

ELA
68

Structured Jordan Canonical Forms




the indenite symmetric matrix  := I0 ,0I , where I is the k  k identity
matrix. We consider the following classes of matrices.
Definition 1.1. Let R and C denote the real and complex eld, respectively.
A matrix C 2 C ( + )( + ) is called  -Hermitian if C  = (C  ) . C is called
 -symmetric if it is  -Hermitian and real.
A matrix C 2 C ( + )( + ) is called  -skew Hermitian if C  = ,(C  ) . C is
called  -skew symmetric if it is  -skew Hermitian and real.
A matrix G 2 C ( + )( + ) is called  -unitary if G  G =  . It is called
 -orthogonal if it is  -unitary and real. Note that the  -Hermitian matrices
form a Jordan algebra, the  -skew Hermitian matrices from a Lie algebra, and the
 -unitary matrices form a Lie group. The algebras and group are invariant under
similarity transformations with  -unitary matrices.

p

p;q

p

q

p

p

q

p

q

p;q

k

q

p;q

p;q

p;q

H

p;q
q

p;q

p;q

p;q

p;q

H

p;q

p

q

p

p;q

q

H

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

Proposition 1.2.

1. If C is  -Hermitian and G is  -unitary then G ,1 CG is  -Hermitian.
2. If C is  -skew Hermitian and G is  -unitary then G ,1 CG is  -skew Hermitian.
3. If G1 and G2 are  -unitary then G1 G2 is also  -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
 -Hermitian, skew Hermitian or unitary, it is dicult to derive the appropriate
structured Schur like forms with similarity transformations that are both unitary and
 -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 Schurlike forms. To make the idea more clear let us consider the case of  -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 matrices. A very simple approach to obtain a canonical form is the idea to express the
 -Hermitian matrix C as an Hermitian pencil  ,  C . Using congruence
transformations U ( ,  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 that U  U =  , 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  -Hermitian matrices and  -skew Hermitian matrices under  -unitary

similarity transformations in Section 3 and Section 4, respectively. By combining the
Cayley transformation and the structured canonical forms for  -skew Hermitian
matrices we will then derive the structured canonical forms for  -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. Here C and R represent
the complex and real case respectively, and J and U represent the classical structured
Jordan canonical forms and the structured Jordan forms under  -unitary similarity
p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

H

p;q

H

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

ELA
69

Volker Mehrmann and Hongguo Xu

C& J
R& J
C& U
R& U

 -Hermitian  -skew Hermitian  -unitary
Theorem 3.1
Theorem 4.1
Theorem 5.9
Theorem 3.2
Theorem 4.2
Theorem 5.10
Theorem 3.3
Theorem 4.3
Theorem 5.11
Theorem 3.6
Theorem 4.4
Theorem 5.12
p;q

p;q

p;q

Main results
Table 1.1

transformations, respectively.
2. Preliminaries. In this section we introduce the notation and give some preliminary results that are needed for the canonical forms. Our construction of structured 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 matrix A. 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 A corresponding to 1 2 (A) and let the columns of V span the right
invariant subspace corresponding to 2 2 (A). If 1 6= 2 then U V = 0 and if
1 = 2 then det(U V ) 6= 0:
Let
3
2
0 1
66
. . . . . . 77
7
6
N := 6
. . . 75
4
1
0
H

H

r

be an r  r nilpotent Jordan block. Dene accordingly
3
3
2
2
1
,1
7
6

77 :
(,1)2
^ := 664
7
;
P
P := 64
5
5




1
(,1)

For any given nilpotent matrix
r

r

r

r

N = diag(N 1 ; : : : ; N s )
r

r

we set

P := diag(P 1 ; : : : ; P s );
N

r

r

P^ := diag(P^ 1 ; : : : ; P^ s ):
N

r

Then these matrices have the following easily veried properties.
Proposition 2.2.

i) P = P ,1 = (,1) ,1 P ;
ii) P ,1 N P = ,N ;
iii) P^ = P^ = P^ ,1 ;
iv) P^ ,1 N P^ = N .
H
r

r
H

N

N

H
N

N

H

N

r

N

N

r

r

r

ELA
70

Structured Jordan Canonical Forms

For matrices A and B , A
B = [a B ] denotes the Kronecker product and for a
t  t matrix Z with t = 1, or 2 we set
N (Z ) := I
Z + N
I ; N (Z ) := I
Z + N
I :
ij

r

r

r

Also set

t



t





1 1 = 10 ,01 ;



J1 = ,01 10 :

;

From [5], or from a direct derivation as done in [8] we have the following properties.
Let C be a complex square matrix
and let  be an eigenvalue

Re

of C with associated Jordan structure N (). If  := , Im Im
Re  , then we have
the following results.
I.a If C is complex 
-Hermitian, then there exists a nonsingular matrix U such
that
i) if Im  6= 0, then



^ 
N
(

)
0
0
P
; C U = U 0 N () ;
U  U = P^
0
Proposition 2.3.

p;q

H

N

p;q

H
N

ii) if  is real, then

U  U = diag(1 P^ 1 : : : ;  P^ s ); C U = UN ();
where all  = 1.
I.b If C is real 
-symmetric, then there exists a real nonsingular matrix U such
H

s

r

p;q

r

i

p;q

that
i) if Im  6= 0, then

U  U = P^
1 1 ;
T

p;q

N

C U = UN ();

;

ii) if  is real, then

U  U = diag(1 P^ 1 : : : ;  P^ s ); C U = UN ();
where all  = 1.
II.a If C is complex 
-skew Hermitian, then there exists a nonsingular matrix U
T

s

r

p;q

r

i

p;q

such that
i) if Re  6= 0, then



0

U  U= P
H

p;q

H
N

P



N

0

;





C U = U N 0() N (0,) ;

ii) if Re  = 0, then

U  U = diag(1 P 1 : : : ;  P s ); C U = UN ();
where  = i for even r and  = 1 if odd r .
II.b If C is real 
-skew symmetric, then there exists a real nonsingular matrix U
H

p;q

k

such that

r

k

p;q

k

s

r

k

ELA
71

Volker Mehrmann and Hongguo Xu

i) if Re  Im  6= 0, then

U T p;q U =






PN
1;1 ; C U = U N ()
0
PNT
1;1
0
0 N (,) ;
0

ii) if Re  6= 0, Im  = 0 then

U T p;q U =




PN ;
PNT 0
0




N
(

)
0
C U = U 0 N (,) ;

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

U T p;q U = diag(Pr1
1 ; : : : ; Prs
s ); C U = UN ((Im )J1 );
where k = k I2 for odd rk and k = (Im k )J1 for even rk , and k is as in the
complex case.
iv) if  = 0, then






U T p;q U = diag 1 P2u1 +1 ; : : : ; a P2ua +1 ; P 0T P20v1 ; : : : ; P0T P20vb ;
2v1
2vb
C U = UN := U diag(N2u1 +1 ; : : : ; N2ua +1 ; N2v1 ; N2v1 ; : : : ; N2vb ; N2vb ):
Note that the parameters k are invariant in the sense that for each group of Jordan
blocks with the same size corresponding to  the 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() = f1 ; : : : ; s g and 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 Proposition 2.3.
Proposition 2.4. Let  be an eigenvalue of a square matrix C . Then we have
the following properties.
 is also an
I. If C is p;q -Hermitian (both complex and real) and Im  6= 0 then 
eigenvalue of C with the same Jordan structure as .
 is
II.a If C is complex p;q -skew Hermitian and  is not purely imaginary, then ,
also an eigenvalue of C with the same Jordan structure as .
, ,, ,
II.b If C is real p;q -skew symmetric and  is not purely imaginary, then 
are eigenvalues of C 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

p 
2

(2.1)

r = 2

for which we have
Hr

(2.2)





(2.3)







0 Ir
Ir 0
Ir 0 r = 0 ,Ir :

For A 2 Cnn a simple calculation yields
,n 1 A0


Ir ,Ir ;
Ir Ir

0



" AAH

2
AH n = , A2AH

#

, A2AHH :
AA
2

ELA
72

Structured Jordan Canonical Forms

A variation of r is
(2.4)

p 2 Ir p0 ,Ir 3
^ r = 2 4 0 2 0 5 :
2

Ir

0

Ir

We will also need, in the following, the symmetric and skew symmetric part of
Jordan blocks
1
1
Nr+ = (Nr + NrT ); Nr, = (Nr , NrT );
2
2
and for a t  t matrix Z with t = 1, 2, we set
Nr+ (Z ) = Ir
Z + Nr+
It ; Nr, (Z ) = Ir
Z + Nr,
It :

Similarly we denote
1
2
and for Z of t  t with t = 1, 2,

1
2

N + = (N + N T ); N , = (N , N T );

N + (Z ) = I
Z + N +
It ; N , (Z ) = I
Z + N ,
It :

Finally the symbol ek 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 canonical forms for p;q -Hermitian matrices. We will always consider two forms, a structured 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. Let C 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(Rc+ ; Rc,; Rr );

where the blocks are
Rc+ = diag(H1 (1 ); : : : ; H ( )); Rc, = diag(H1 (1 ); : : : ; H ( ));
Rr = diag(M1 (1 ); : : : ; M ( ));

with substructures
Hk (k ) = k I + Hk ; Hk (k ) = k I + Hk ; Hk = diag(Np 1 ; : : : ; Np
k;

for k = 1; : : : ; , and
Mk (k ) = k I + Mk ; Mk = diag(Nq 1 ; : : : Nq
k;

for k = 1; : : : ;  .

k;tk

);

k;sk

);

ELA
Volker Mehrmann and Hongguo Xu

The matrix U satises

2

U H p;q U

(3.1)

0

= 4 WcH
0

Wc

0
0

0
0

Wr

73

3
5;

with Wc = diag(P^H1 ; : : : ; P^H ) and Wr = diag(W1r ; : : : ; Wr ), where for k = 1; : : : ; 
we have P^Hk = diag(P^pk;1 ; : : : ; P^pk;sk ) and for k = 1; : : : ;  and Ind(k ) = fk;1 , : : :,
k;tk g we have Wkr = diag(k;1 P^qk;1 ; : : : ; k;tk P^qk;tk ).
Proof. For each nonreal eigenvalue k with the corresponding Jordan structure
Hk (k ), by I.a, i) of Proposition 2.3, we can choose a matrix Uk such that

^Hk 
0
P
H
(3.2)
Uk p;q Uk = ^ H
; C Uk = Uk diag(Hk (k ); Hk (k )):
PHk 0

Partition Uk = [Uk;1 ; Uk;2 ], where Uk;1 , Uk;2 have the same size and set
Uc

= [U1;1; : : : ; U;1 ; U1;2 ; : : : ; U;2 ] = [U1c ; U2c ]:

Note that by the symmetry of C , the columns of U1c , p;q U2c and U2c , p;q U1c form
bases of the right and left invariant subspaces corresponding to the two disjoint sets
of eigenvalues f1 ; : : : ;  g and f1 ; : : : ;  g, respectively. By Proposition 2.1 and
(3.2) we have
UcH p;q Uc





= W0H W0c ;
c

CU

= U diag(Rc+ ; Rc,);

with Wc , Rc+ and Rc, as asserted.
For each real eigenvalue k with the corresponding Jordan structure Mk (k ), by
I.a, ii) of Proposition 2.3 we can choose a matrix Vk such that
VkH p;q Vk = diag(k;1 P^qk;1 ; : : : ; k;tk P^qk;tk );
where Ind(k ) = fk;1 ; : : : ; k;tk g and C Vk = Vk Mk (k ). Set Ur = [V1 ; : : : ; V ], then
by Proposition 2.1 we have
UrH p;q Ur

= Wr ;

CUr

= Ur Rr ;

where Wr and Rr are of the asserted forms and with U = [Uc ; Ur ] 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 ; : : : ;  and pairwise dierent eigenvalues 1 ; : : : ;  , with positive
imaginary parts.
Then there exists a real full rank matrix U such that
U ,1 CU

= diag(Rc ; Rr );

where
Rc = diag(H1 (1 ); : : : ; H ( ))

ELA
74

Structured Jordan Canonical Forms

and for k = 1; : : : ;  the subblocks
are Hk (k ) = diag(Npk;1 (k ); : : : ; Npk;sk (k )) with

k Im k
k = ,Re
Im  Re  . The other diagonal block is
k

k

Rr = diag(M1 (1 ); : : : ; M ( ));

where for k = 1; : : : ;  the subblocks are Mk (k ) = k I + Mk with Mk = diag(Nqk;1 ,
: : :, Nqk;tk ). The matrix U has the form

U

(3.3)

T

p;q

U=



Wc

0

0



Wr ;

where Wc = diag(P^H1
1;1 ; : : : ; P^H
1;1 ), Wr = diag(W1r ; : : : ; Wr ), and where for
k = 1; : : : ;  we have P^Hk = diag(P^pk;1 ; : : : ; P^pk;sk ) and for Ind(k ) = fk;1 ; : : : ; k;tk g
and k = 1; : : : ;  we have Wkr = diag(k;1 P^qk;1 ; : : : ; k;tk P^qk;tk ).
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 ; : : : ;  and pairwise dierent eigenvalues 1 ; : : : ;  , with positive
imaginary parts. Then there exists a p;q -unitary matrix U such that
2

Rc

Tc

3

+
7
6
U ,1 CU = 64 ,T H Rr Rc Tr 75 :
c
Rr,
,TrH

(3.4)

For the blocks we have the following substructures.
i) The blocks with index c, associated with the nonreal eigenvalues, are
Rc = diag(R1c ; : : : ; Rc ); Tc = diag(T1c; : : : ; Tc);

where for k = 1; : : : ;  we have
Rkc = diag(Np+k;1 (Re k ); : : : ; Np+k;sk (Re k ));
Tkc = , diag(Np,k;1 (i Im k ); : : : ; Np,k;sk (i Im k )):

ii) The blocks with index r, associated with the real eigenvalues are
Rr+ = diag(C1 ; : : : ; C ); Rr, = diag(D1 ; : : : ; D ); Tr = diag(F1 ; : : : ; F ):

For k = 1; : : : ;  these have the substructures
Ck = diag(Cke ; Ck+ ; Ck, ); Dk = diag(Dke ; Dk+ ; Dk, ); Fk = diag(Fke ; Fk+ ; Fk, );

where

1
1
+
H
Cke = diag(Nq+k;1 (k ) + k;1 eqk;1 eH
qk;1 ; : : : ; Nqk;tk (k ) + k;tk eqk;tk eqk;tk );
2

2

ELA
75

Volker Mehrmann and Hongguo Xu

Dke = diag(Nq+ 1 (k ) , 12 k;1 eq 1 eHq 1 ; : : : ; Nq+ (k ) , 12 k;t eq eHq );
Fke = diag(,Nq, 1 + 12 k;1 eq 1 eHq 1 ; : : : ; ,Nq, + 21 k;t eq eHq );
p
#
"
#!
"
+ (k ) 2 eu
+ (k ) p2 eu 1
N
N
u
u
2
+
2
1
p2 H
;
Ck = diag
; : : : ; p2 H
k
k
2 eu 1
2 eu
Dk+ = diag(Nu+ 1 (k ); : : : ; Nu+ (k ));
#
"
#!
"
,p Nu,
,p Nu, 1
+
; : : : ; 2 eH
;
Fk = diag
2 H
2 eu 1
2 u
Ck, = diag(Nv+ 1 (k ); : : : ; Nv+ (k ));
p2 H #!
"
"
p2 H #

,

,
e
k
k
v
,
2 ev
2
1
p2
; : : : ; p2
;
Dk = diag
+
+
, 2 ev
Nv (k )
, 2 ev 1 Nv 1 (k )
k;

k;

k;

k;

k;

k;

k

k;tk

k;

k;

k

k;tk

k;tk

k;tk

k;tk

k;wk

k;wk

k;

k;tk

k;wk

k;

k;wk

k;

k;wk

k;

k;wk

k;

k;zk

k;zk

k;

p

k;

p

k;

Fk, = diag([ 22 ev 1 ; ,Nv, 1 ]; : : : ; [ 22 ev
k;

k;zk

k;

 

k;zk

k;zk

; ,Nv, ]):
k;zk

s

p ;:::;p

Here each nonreal k ( k ) has k Jordan blocks of sizes k;1
k;sk and each
real eigenvalue k has
a) k even sized Jordan blocks of sizes
k;1
k;tk and the corresponding structure
inertia indices k;1
k;tk ;
b) k odd sized Jordan blocks of sizes
, corresponding to the
k;1
k;wk
structure inertia index ;
c) k odd sized Jordan blocks of sizes
, corresponding to the
k;1
k;zk
structure inertia index
.
P Psi
Proof
1

i=1 j =1 ij

t

w



2q ; : : : ; 2q

 ;:::;
1

z

. Let m :=
matrix Uc such that

,1

2u + 1; : : : ; 2u

+1

2v

+1

+ 1; : : : ; 2v

p . For  ; : : : ;  , by Theorem 3.1 there exists a









+
U^cH p;q U^c = W0H W0c ; C U^c = U^c R0c R0, :
c
c
Note WcH Wc = Wc2 = Im . Setting U~c := U^c diag(Im ; Wc ) then using the form of Rc,

and Proposition 2.2 we have








+

U~cH p;q U~c = I0m I0m ; C U~c = U~c R0c (R+0 )H :
c
Now let Uc := U~c m , where m is dened in (2.1). By (2.2) and (2.3) and the special
form of Rc+ we then get
(3.5)

UcH p;q Uc = m;m;





C U = U ,RTcH RTcc ;
c

where Rc and Tc are in the asserted forms.
For real eigenvalues 1 ; : : : ;  the situation is relatively complicated. In this case
we have to transform the Jordan blocks one by one. Let  be a real eigenvalue of C
and Nr () be a Jordan block. Following from Proposition 2.3 there exists a matrix
U^ such that
^ r ();
U^ H p;q U^ = P^r ; C U^ = UN

ELA
76

Structured Jordan Canonical Forms

where  = 1. When r is even let U := U^ diag(I 2 ; P^2 ) 2 then one can verify that
r

r

(3.6) U  U =  2 2 ; C U = U
H

r

p;q

"

r

r

Similarly when r is odd setting U := U^ diag(I
in (2.4), then if  = 1,
U  U=
H

"

p;q

r

2

U  U=

"

p;q

2
6
C U = U 64

(3.8)

r

,1
2

r

2

r

2

#

;
p2
N +,1 ()
2 e ,2 1
p22
, 2 e ,2 1 p 
,N ,,1 , 22 e ,2 1
+1
2

,N ,,2 1 3
p
7
, 22 e ,2 1 75 :

r

r

r

H

H

r

r

is dened

7
7;
5

r

r

r

3

r

H

,I

0

H

#

r

0

r

r

r

H

I ,2 1
r

r

r

r

H

H

H

r

r

^ r,1 ; where ^ r,1
+1 ;  P^r,1 )
2
2
2
2

r

6

and if  = ,1

r

r

;
p
N +,1 () 22 e ,2 1 ,N ,,1
p2 2
p22

2 e ,2 1
2 e ,2 1
p
,N ,,1 , 22 e ,2 1 N +,1 ()

,I

0

C U = U 64

(3.7)

0

I +1
2

H

r

r

#

,N ,2 + 12 e 2 e 2
N +2 () , 12 e 2 e 2 :

N +2 () + 12 e 2 e 2
,N ,2 , 12 e 2 e 2

r

;

r

r

N +,1 ()

r

2

r

2

Let  be a real eigenvalue with associated even sized Jordan blocks of sizes 2q 1 ,
: : :, 2q , and associated odd sized Jordan blocks of sizes 2u 1 + 1; : : : ; 2u
+1
and 2v 1 + 1, : : :, 2v + 1 corresponding to the structure inertia indices 1 and ,1
respectively. By Proposition 2.3 there exists a matrix U^ such that
k

k;

k;tk

k;

k;

k;wk

k;zk

k

U^  U^ = diag( 1 P^2 1 ; : : : ;  P^2 ; P^2 1 +1 ; : : : ; P^2
+1 ;
,P^2 1 +1 ; : : : ; ,P^2 +1 );
C U^ = U^ diag(N2 1 ( ); : : : ; N2 ( ); N2 1 +1 ( );
: : : ; N2
+1 ( ); N2 1 +1 ( ); : : : ; N2
+1 ( )):
H
k

p;q

k

k;

qk;

k;tk

qk;t

uk;

k

vk;

k

k

qk;

k

vk;z

qk;t

uk;w

k

k

uk;w

k

k

vk;

k

k

uk;

k

k

uk;z

k

k

Set
U~ := U^ diag(Z 1 ; : : : ; Z
k

e
k;

k

e
k;tk

; Z +1 ; : : : ; Z + ; Z ,1 ; : : : ; Z , );
k;

k;wk

k;

k;zk

where
Z = diag(I
Z , = diag(I
e
k;j

qk;j

k;j

vk;j

;  P^ ) ; Z + = diag(I
^ :
+1 ; ,P^ )
k;j

qk;j

qk;j

vk;j

k;j

uk;j

+1 ; P^

uk;j

)^

uk;j

vk;j

Partition
U~ = [V 1 ; W 1 ; : : : ; V
k

e
k;

e
k;

; W ; V +1 ; W +1 ; : : : ; V + ; W + ;
V ,1 ; W ,1 ; : : : ; V , ; W , ];

e
k;tk

k;

e
k;tk

k;

k;

k;

k;zk

k;wk

k;zk

k;wk

;

ELA
77

Volker Mehrmann and Hongguo Xu

e ; W e are qk;j , the columns of V + , W + , V , , W , are
where the columns of Vk;j
k;j
k;j k;j k;j k;j
uk;j + 1, uk;j , vk;j , vk+1 + 1, respectively. Set
e ; V + ; : : : ; V + ; V , ; : : : ; V , ];
Vk = [Vk;e 1 ; : : : ; Vk;t
k;1
k;w
k;1
k;z
+
+
,
, ]
e
e
Wk = [Wk;1 ; : : : ; Wk;t ; Wk;1 ; : : : ; Wk;w ; Wk;1 ; : : : ; Wk;z
and Uk = [Vk ; Wk ]. Then by employing (3.6) { (3.8) we have




Fk ;
; C Uk = Uk ,CFkH D
UkH p;q Uk = In0 1 ,I0
k
n 2
k
k

k

k

k

k

k

k;

k;

Pk

P

P

t
w u + z v and
where Ck , FP
k , Dk are asPasserted, nk;P1 = wk + l=1 qk;l +
l=1 k;l
lP
=1 k;l
P
nk;2 = zk + tl=1 qk;l + wl=1 uk;l + zl=1 vk;l . Set n1 = k=1 nk;1 , n2 = k=1 nk;2 .
Then with
k

k

(3.9)

UrH p;q Ur =



k

k

Vr = [V1 ; : : : ; V ];
and Ur = [Vr ; Wr ] we have

k

In1
0

W r = [W 1 ; : : : ; W  ] ;


0

,In2





+
; CUr = Ur ,RTrH RT,r ;
r
r

Finally set U = [Vc ; Vr ; Wc ; Wr ], then by Proposition 2.1 and by above construction we have


U H p;q U = Im0+n1 ,Im0+n :
2
Since U is nonsingular it follows that U H p;q U is congruent to p;q and hence m+n1 =
p, m + n2 = q and U H p;q U = p;q . Equation (3.4) then follows from (3.5) and (3.9).
Remark 3.4. The dierence between the structured canonical forms of Theorems 3.1 and 3.3 is that in order to get a p;q -unitary transformation matrix we
need to rene 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
tk
X

k=1

X

j =1
t
X

k=1
k=1 j =1

X
jp , qj = j (wk , zk )j:
k=1

j =1

q=

(3.10)

 X
sk
X


X

p=

k=1 j =1
 X
s
X
k

pk;j +
pk;j +

(wk +
(zk +

k

qk;j +

qk;j +

wk
X

j =1
w
X
k

j =1

uk;j +

uk;j +

zk
X

j =1
z
X
k

j =1

vk;j );

vk;j );

These relationship show that the parameters p; q will aect the eigenstructure of C .
For example, we get in the case p = 0 (or q = 0) that C , which is Hermitian now,

ELA
78

Structured Jordan Canonical Forms

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
2 minfp; 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 if jp , qj 6= 0, then C
must have real eigenvalues with at least jp , qj odd 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 ; : : : ;  and pairwise dierent eigenvalues 1 ; : : : ;  with positive
imaginary parts. Then there exists a real p;q -orthogonal matrix U , such that
2

Rc+

3

Tc

+
7
6
U ,1 CU = 64 ,T T Rr R, Tr 75 :
c
c
,TrT
Rr,
i) The blocks with index c, associated with nonreal eigenvalues, are
Rc+ = diag(A1 ; : : : ; A ); Rc, = diag(B1 ; : : : ; B ); Tc = diag(T1c; : : : ; Tc);
where for k = 1; : : : ;  we have
Ak = diag(Aek ; Aok ); Bk = diag(Bke ; Bko ); Tkc = diag(Tke ; Tko);
Aek = diag(Np+ 1 ((Re k )I2 ) + Ek;1 ; : : : ; Np+ ((Re k )I2 ) + Ek;s );
Bke = diag(Np+ 1 ((Re k )I2 ) , Ek;1 ; : : : ; Np+ ((Re k )I2 ) , Ek;s );
Tke = , diag(Np, 1 ((Im k )J1 ) , Ek;1 ; : : : ; Np, ((Im k )J1 ) , Ek;s );
p
#
"
Nl+p1 ((Re k )I2 ) 22 e2l 1 ,1 ;
o
Ak = diag
2 T
Re k
2 e2l 1 ,1
p
" +
#!
Nl p ((Re k )I2 ) 22 e2l ,1
:::;
;
2 T
Re k
2 e2l ,1
p
"
#
Nl+ 1p((Re k )I2 ) 22 e2l 1
o
Bk = diag
;
2 T
Re k
2 e2 l 1
p
#!
" +
Nl p ((Re k )I2 ) 22 e2l
;
:::;
2 T
Re k
2 e2 l
"
, ((Im k )J1 ) , p2 e2l #
,
N
1
lp1
2
Tko = diag
;
2 eT
, Im k
2 2l 1 ,1
p
"
#!
,Nl,p ((Im k )J1 ) , 22 e2l
:::;
;
2 T
, Im k
2 e2l ,1

(3.11)

k;

k;sk

k

k;

k;sk

k

k;

k

k;sk

k;

k;

k;

k;xk

k;xk

k;xk

k;

k;

k;

k;xk

k;xk

k;xk

k;

k;

k;

k;xk

k;xk





k;xk

with Ek;j = 21 00 0 .
1;1
ii) The blocks with index r, associated with real eigenvalues, are

Rr+ = diag(C1 ; : : : ; C ); Rr, = diag(D1 ; : : : ; D ); Tr = diag(F1 ; : : : ; F ):

ELA
79

Volker Mehrmann and Hongguo Xu

k = 1; : : : ;  the substructures
Ck = diag(Cke ; Ck+ ; Ck, ); Dk = diag(Dke ; Dk+ ; Dk, ); Fk = diag(Fke ; Fk+ ; Fk, );

These have for

where

1
1
Cke = diag(Nq+ 1 (k ) + k;1 eq 1 eTq 1 ; : : : ; Nq+ (k ) + k;t eq

eTq );
2
2
1
1
Dke = diag(Nq+ 1 (k ) , k;1 eq 1 eTq 1 ; : : : ; Nq+ (k ) , k;t eq eTq );
2
2
1
1
Fke = diag(,Nq, 1 + k;1 eq 1 eTq 1 ; : : : ; ,Nq, + k;t eq eTq );
2
2
p
#!
"
#
"
p2
+
+
Npu (k ) 22 eu
Npu 1 (k ) 2 eu 1
+
Ck = diag
;:::;
;
2 T
2 T
k;

k;

k;

k;tk

k

k;tk

k;tk

k;

k;

k;

k;tk

k

k;tk

k;tk

k;tk

k;tk

k;

k;

k;

k;

k;

k

k;tk

k;wk

k;wk

k
2 eu 1
2 eu
+
+
+
Dk = diag(Nu 1 (k ); : : : ; Nu (k ));
"
#
"
#!
,p Nu,
,p Nu, 1
+
Fk = diag
;:::;
;
2 T
2 T
2 eu 1
2 eu
Ck, = diag(Nv+ 1 (k ); : : : ; Nv+ (k ));
"
"
p2 T #
e

,
k
v
,
2
1
p2
p2k
;:::;
Dk = diag
+
, 2 ev 1 Nv 1 (k )
, 2 ev
k;

k

k;wk

k;

k;wk

k;

k;wk

k;

k;wk

k;

p

k;zk

, 22 eTv

k;zk

k;

Fk, = diag([

p

k;

p

k;

2
2
,
2 ev 1 ; ,Nv 1 ]; : : : [ 2 ev
k;

k;

k;zk

k;zk

Nv+ (k )

#!

;

k;zk

; ,Nv, ]):
k;zk

2

2

Each k (k ) has sk even sized Jordan blocks of sizes pk;1 ; : : : ; pk;sk , and xk odd
sized Jordan blocks of sizes lk;1
; : : : ; lk;xk .
For each real eigenvalue k there are
a) tk even sized Jordan blocks of sizes qk;1 ; : : : ; qk;tk corresponding to the structure
inertia indices k;1 ; : : : ; k;tk ;
b) wk odd sized Jordan blocks of sizes uk;1
; : : : ; uk;wk
corresponding to the
structure inertia index ;
c) zk odd sized Jordan blocks of sizes vk;1
; : : : ; vk;zk
corresponding to the
structure inertia index
.
Proof
1 ; : : : ;  

2

+1

2

+1

2

1

,1

2

2

+1

2

+1

2

+1

2

+1

. For real eigenvalues
, using I.b, ii) of Proposition 2.3, as in the
proof of Theorem 3.3, there exists a real matrix Ur := [Vr ; Wr ] such that

U p;q Ur =
T
r



In1

0

0



,In2 ; CUr = Ur





Rr+ Tr ;
,TrT Rr,

which is the real version of (3.9).
For a nonreal eigenvalue  of C with a Jordan block Nr (), by Proposition 2.3
there is a real matrix U^ such that
^ ();
C U^ = UN
U^ T p;q U^ = P^r
1;1 ;




 Im 
where  = ,Re
Im  Re  : As for (3.6){(3.8), if r is even set
U := U^ diag(Ir ; P^ 2
r


1;1 )r ;

ELA
80

Structured Jordan Canonical Forms

and if r is odd then set

2 p2

2 Ir,1 0
6
0
1
6

U := U^ diag(Ir+1 ; P^r,2 1
1;1 ) 4

0
p2 0
I
0
r
,
1
2

p

, 22 Ir,1 0
0
p2 0

3

0 77 :
,1 5
0

2 Ir,1

By Proposition 2.1 we have that
(P^s
1;1 ),1 = P^s
1;1 = (P^s
1;1 )T ; (P^s
1;1)T (Ns ())(P^s
1;1 ) = (Ns ())T ;
and some simple calculations yield

U T p;q U =









A T
0 ,Ir ; C U = U ,T T B ;

Ir

0

where, if r = 2s, then

A = Ns+ ((Re )I2 ) + Er ; B = Ns+ ((Re )I2 ) , Er ; T = ,Ns, ((Im )J1 ) + Er ;
with Er = 12



"







0 0 ; and if r = 2s + 1, then with J = 0 1 ,
1
,1 0
0 1;1

+
)I2 )
A = Ns p((Re
2 eT
2 r,2
"
,
)J1 )
T = ,Nsp((Im
2 eT
2 r,2

#
"
#
p2
+ ((Re )I2 ) p2 er,1
N
e
r
,
2
2
2
; B = s p2 T
;
Re 
er , 1
Re 
2
#
p
, 22 er,1 :

, Im 

Now as for the case of real eigenvalues in the proof for Theorem 3.3, for nonreal
eigenvalues 1 ; : : : ;  there exists a real matrix Uc := [Vc ; Wc ] such that

UcT p;q Uc = m;m ;





+
CUc = Uc ,RTc T RT,c ;
c
c
P

P

P

k
k
where Rc+, Tc, Rc, are in the asserted forms and m = k=1 ( sj=1
2pkj + xj=1
(2lkj +
1)).
Set U = [Vc ; Vr ; Wc ; Wr ] then analogously we get that U is real p;q -orthogonal
and U ,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 Jordan 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 Proposition 2.3 we have the following Jordan canonical forms both for complex and real
p;q -skew Hermitian matrices.

ELA
Volker Mehrmann and Hongguo Xu

81

Theorem 4.1. Let C be a p;q -skew Hermitian matrix with pairwise dierent
purely imaginary eigenvalues 1 ; : : : ;  and pairwise dierent eigenvalues 1 ; : : : ; 
with positive real parts. Then there exists a nonsingular matrix U such that

U ,1 CU = diag(Rc+ ; Rc,; Rg );

i) The diagonal blocks with index c, associated with eigenvalues not on the imaginary
axis, are
Rc+ = diag(H1 (1 ); : : : ; H ( )); Rc, = diag(H1 (,1 ); : : : ; H (, ));

where for k = 1; : : : ;  we have
Hk (k ) = k I + Hk ; Hk (,k ) = ,k I + Hk ; Hk = diag(Npk;1 ; : : : ; Npk;sk ):

ii) The block Rg , associated with purely imaginary eigenvalues, has the form
Rg = diag(M1 (1 ); : : : ; M ( ));

where Mk (k ) = k I + Mk and for k = 1; : : : ;  we have Mk = diag(Nqk;1 , : : :, Nqk;tk ).
The matrix U has the form
2

0

Wc

0

0

U H p;q U = 4 WcH 0

0
0

Wg

3

5;

where
Wc = diag(PH1 ; : : : ; PH );

Wg = diag(W1g ; : : : ; Wg );

and for k = 1; : : : ;  we have PHk = diag(Ppk;1 ; : : : ; Ppk;sk ), and with Ind(k ) =
fk;1 ; : : : ; k;tk g for k = 1; : : : ;  we have Wkg = diag(k;1 Pqk;1 ; : : : ; k;tk Pqk;tk ).
Theorem 4.2. Let C be a real p;q -skew symmetric matrix with pairwise different 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 . Furthermore 0 may be an eigenvalue.) Then there exists a
real nonsingular matrix U such that

U ,1 CU = diag(Rc+ ; Rc,; Rg ):

i) The blocks with index c, associated with eigenvalues not on the imaginary axis, are
Rc+ = diag(R^c+ ; R~c+ ), with R^c+ = diag(K1 (1 ); : : : ; K ( )) where for k = 1; : : : ; 
we have Kk (k ) = k I + Kk and Kk = diag(Nfk;1 ; : : : ; Nfk;lk ). Analogously R~c+ =
k Im k
diag(H1 (1 ); : : : ; H ( )), where for k = 1; : : : ;  we have k = ,Re
Im k Re k
and Hk (k ) = diag(Npk;1 (k ); : : : ; Npk;sk (k )).
The block Rc, = diag(R^c, ; R~c, ), has the same substructure as Rc+ just replacing
j with ,j and j by ,j .
ii) The block Rg , associated with purely imaginary eigenvalues, has the structure
Rg = diag(M1 ((Im 1 )J1 ); : : : ; M ((Im  )J1 ); M0 );

ELA
82

Structured Jordan Canonical Forms

where for k = 1; : : : ;  we have
Mk ((Im k )J1 ) = diag(Nqk;1 ((Im k )J1 ); : : : ; Nqk;tk ((Im k )J1 ));

and where
M0 = diag(N2g1 +1 ; : : : ; N2ga +1 ; N2h1 ; N2h1 ; : : : ; N2hb ; N2hb )

is the structure associated with the eigenvalue 0.
The matrix U has the form
2
0 Wc 0
U T p;q U = 4 WcT 0 0
0 0 Wg
where Wc = diag(W^ c ; W~ c ) with

3
5;

~ c = diag(PH1
1;1 ; : : : ; PH
1;1 );
W

^ c = diag(PK1 ; : : : ; PK );
W
and where

PKk = diag(Pfk;1 ; : : : ; Pfk;lk );

PHk = diag(Ppk;1 ; : : : ; Ppk;sk ):

The block Wg has the form Wg = diag(W1g ; : : : ; Wg ; W0 ), where for k = 1; : : : ;  and
Ind(k ) = fk;1 ; : : : ; k;tk g we have Wkg = diag(Pqk;1
k;1 ; : : : ; Pqk;tk
k;tk ), with
k;j = k;j I2 if qk;j is odd and k;j = (Im k;j )J1 if qk;j is even.
Finally for Ind(0) = f10 ; : : : ; a0 ; i; ,i; : : : ; i; ,ig we have


0 P2h1
W0 = diag(10 P2g1 +1 ; : : : ; a0 P2ga +1 ; P T
2 h1 0



;:::;



0

P2Thb

P2hb



0

):

After determining the Jordan structure under non p;q -unitary similarity transformations 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 a p;q -unitary matrix U , such that
2

(4.1)

6

U ,1 CU = 64

Rc
TcH

Rg+
TgH

Tc
Rc

3

Tg 7
7
Rg,

5:

For the dierent blocks we have the following substructures.
i) The blocks with index c, associated with eigenvalues not on the imaginary axis, are
Rc = diag(R1c ; : : : ; Rc ) and Tc = diag(T1c ; : : : ; Tc ) where for k = 1; : : : ; 
Rkc = diag(Np,k;1 (i Im k ); : : : ; Np,k;sk (i Im k ));
Tkc = , diag(Np+k;1 (Re k ); : : : ; Np+k;sk (Re k )):

ii) The blocks with index g, associated with purely imaginary eigenvalues, are

Rg+ = diag(C1 ; : : : ; C ); Rg, = diag(D1 ; : : : ; D ); Tg = diag(F1 ; : : : ; F );

ELA
83

Volker Mehrmann and Hongguo Xu

where for k = 1; : : : ;  the substructures are
Ck = diag(Cke ; Ck+ ; Ck, ); Dk = diag(Dke ; Dk+ ; Dk, ); Fk = diag(Fke ; Fk+ ; Fk,);
with further partitioning
Cke = diag(Nq, 1 (k ) + 12 ik;1 eq 1 eHq 1 ; : : : ; Nq, (k ) + 21 ik;t eq eHq );
Dke = diag(Nq, 1 (k ) , 21 ik;1 eq 1 eHq 1 ; : : : ; Nq, (k ) , 21 ik;t eq eHq );
Fke = diag(,Nq+ 1 + 21 ik;1 eq 1 eHq 1 ; : : : ; ,Nq+ + 12 ik;t eq eHq );
#!
"
#
"
, (k ) p2 eu
, (k ) p2 eu 1
N
N
u
u
+
2
2
p2 H
p21 H
Ck = diag
;:::;
;
k;

k;

k;

k;tk

k

k;tk

k;tk

k;

k;

k;

k;tk

k

k;tk

k;tk

k;

k;

k;

, 2 eu

k;

k

k;1

k

k;tk

k;

k;tk

k;wk

k;wk

, 2 eu

k

k;wk

Dk+ = diag(Nu, 1 (k ); : : : ; Nu, (k ));
"
"
#!
+
+ #
N
N
u
u
1
+
Fk = , diag p2 eH ; : : : ; p2 eH
;
2 u 1
2 u
Ck, = diag(Nv, 1 (k ); : : : ; Nv, (k ));
"
"
p2 H #
e

k
2 v 1 ;:::;
p2
p2k
Dk, = diag
,
, ev N (k )
, 2 ev
p 2 1 v1 p
Fk, = diag([ 22 ev 1 ; ,Nv+ 1 ]; : : : [ 22 ev ; ,Nv+ ]):
k;

k;tk

k;wk

k;wk

k;

k;

k;wk

k;

k;zk

k;zk

k;

k;

k;

k;zk

k;

k;zk

k;

p2 H
#!
e
v
2
;
Nv, (k )
k;zk

k;zk

Each k (,k ) has sk Jordan blocks of sizes pk;1 ; : : : ; pk;s . Each purely imaginary
eigenvalue k has
a) tk even sized Jordan blocks of sizes 2qk;1 ; : : : ; 2qk;t with the corresponding structure
inertia indices i(,1)q 1 +1 k;1 ; : : : ; i(,1)q +1 k;t ;
b) wk odd sized Jordan blocks of sizes 2uk;1 + 1; : : : ; 2uk;w + 1 corresponding to the
structure inertia indices (,1)u 1 +1 ; : : : ; (,1)u +1 ;
c) zk odd sized Jordan blocks of sizes 2vk;1 + 1; : : : ; 2vk;z + 1 corresponding to the
structure indices (,1)v 1 ; : : : ; (,1)v .
Proof. For all eigenvalues 1 ; : : : ;  , by Theorem 4.1 there is a matrix U^c such
that
 +



U^cH p;q U^ = W0H W0c ; C U^c = U^c R0c R0, ;
c
c
,1
^
where Wc , Rc+ , Rc, are dened in Theorem
P Ps4.1. Let Uc := Uc diag(Im ; Wc )m , where
m is dened in (2.1) and m = k=1 j=1 pkj . By using i), ii) of Proposition 2.2
and (2.2), (2.3) we have
k

k

k;tk

k;

k

k

k;wk

k;

k

k;

k;zk

k

(4.2)

UcH p;q Uc = m;m;





CUc = Uc TRHc RTcc ;
c

where Rc , 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 block Nr () with  purely
imaginary. According to Proposition 2.3 for this block there is a matrix U^ such that
^ r ();
U^ H p;q U^ = Pr ; C U^ = UN

ELA
84

Structured Jordan Canonical Forms

where  = 1 if r is odd and  = i if r is even. Similarly if r is even let U :=
U^ diag(I 2 ; (P 2 ),1 ) 2 . Then
r

r

r

"

#
N ,2 () + 21 ie 2 eH2 ,N +2 + 21 ie 2 eH2
U p;q U = 0 ,I ; C U = U ,N + , 1 ie eH N , () , 1 ie eH ;
2 2 2
2 2 2
2
2
2
where  = (,1) 2 i. If r is odd let U := U^ diag(I +1 ; (P ,1 ),1 )^ ,1 . Then


H



0

I2
r

r

r

r

r

r

r

r

r

r

r

r

r

r

r

r

r

2

r

2

2

p2
2
N ,,1 ()
2 e ,2 1
0 3
I ,2 1 0
p22 H
6

0 5 ; C U = U 64 , 2 e ,2 1
U H p;q U = 4 0 
p2
0 0 ,I ,2 1
+
,N ,1 , 2 e ,2 1
2

r

r

r

r

r

r

+1
2 .

r

2

,pN +,2 1

3

r

, 22 eH,2 1 775 ;
r

N ,,1 ()
r

2

where  = (,1)
Note that  = 1, so here U H p;q U is either  +1
,1 or
2 ; 2
 ,2 1 ; +1
depending
on
the
sign
of

.
2
Applying these formulas to all purely imaginary eigenvalues 1 ; : : : ;  , analogous to the real eigenvalue case in Theorem 3.3 for p;q -Hermitian matrices we can
construct a matrix Ug such that
r

r

r

r

r

UgH p;q Ug = n1 ;n2 ;



 +
CUg = Ug TRHg RT,g ;
g

g

where Rg+, Rg, , Tg are dened in the theorem and n1 , n2 are the sizes of Rg+ , Rg, ,
respectively.
The p;q -unitary matrix U can then be generated from Uc, Ug , 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 distinct 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
,1 ; : : : ; , , 1 ; : : : ;  , ,1 ; : : : ; , , ,1 , : : :, , and ,1 ; : : : ; , and also
0 may be another eigenvalue.)
Then there exists a real p;q -orthogonal matrix U , such that
2

Rc

3

Tc

+
6
U ,1 CU = 64 T T Rg Rc Tg
c

(4.3)

TgT

Rg,

7
7;
5

where the dierent blocks have the following substructures:
i) The blocks with index c, associated with the eigenvalues with nonzero real part, are
Rc = diag(R^c ; R~c ); Tc = diag(T^c; T~c );
R^c = diag(R^1c ; : : : ; R^c ); R~c = diag(R~1c ; : : : ; R~c );
T^c = diag(T^1c; : : : ; T^c); T~c = diag(T~1c; : : : ; T~c);
where for k = 1; : : : ;  the substructures are
R^kc = diag(Nf, 1 ; : : : ; Nf, ); T^kc = , diag(Nf+ 1 (k ); : : : ; Nf+ (k ));
k;

k;lk

k;

k;lk

ELA
85

Volker Mehrmann and Hongguo Xu

and for k = 1; : : : ; ,
R~ kc = diag(Np, 1 ((Im k )J1 ); : : : ; Np, ((Im k )J1 ));
T~kc = , diag(Np+ 1 ((Re k )I2 ); : : : ; Np+ ((Re k )I2 )):
k;

k;sk

k;

k;sk

ii) The blocks with index g, associated with the purely imaginary eigenvalues, are
Rg+ = diag(C1 ; : : : ; C ; C0 ); Rg, = diag(D1 ; : : : ; D ; D0 ); Tg = diag(F1 ; : : : ; F ; F0 );
with the partitioning
Ck = diag(Cke ; Ck+ ; Ck, ); Dk = diag(Dke ; Dk+ ; Dk, ); Fk = diag(Fke ; Fk+ ; Fk, );
and for k = 1; : : : ;  the blocks have the further substructure
Cke = diag(Nq, 1 ((Im k )J1 ) + Ek;1 : : : ; Nq, ((Im k )J1 ) + Ek;t );
Dke = diag(Nq, 1 ((Im k )J1 ) , Ek;1 ; : : : ; Nq, ((Im k )J1 ) , Ek;t );
Fke = diag(,Nq+ 1 (02 ) + Ek;1 ; : : : ; ,Nq+ (02 ) + Ek;t );
02
3
0
,
p2
7
Ck+ = diag B
@64 Nu 1 ((Impk )J1)
2 I2 5 ;
(Im k )J1
0 , 22 I2
2
31
0
,
p
6 Nu ((Im k )J1)
7C
2
k;

k

k;tk

k;

k

k;tk

k;

k

k;tk

k;

:::;4

2 I2 5A ;
p2
(Im k )J1
0 , 2 I2
+
,
,
Dk = diag(Nu 1 ((Im k )J1 ); : : : ; Nu ((Im k )J1 ));
# " +
" +
#!
Nu (02 )
Nu p
(0 )
+
1 2
p
;:::;
Fk = , diag
;
0 22 I2
0 22 I2
Ck, = diag(Nv, 1 ((Im k )J1 ); : : : ; Nv, ((Im k )J1 ));
02
3
p2
(Im k )J1
I2
0
2
75 ;
Dk, = diag B
@64 p0
,
N
((Im

)
J
)
k 1
v 1
, 22 I2
31
2
p2
(Im k )J1
I
0
2
2
75CA ;
:::; 6
4 p0
,
, 22 I2 Nv ((Im k )J1 )
 0
  0
p2
p2
Fk, = diag
,
Nv+ 1 (02 ) ; : : : ;
,Nv+ (02 )
I
I
2
2
2
2
k;wk

k;

k;wk

k;wk

k;

k;

k;zk

k;

k;zk

k;

0



k;zk



:

0
Here for j = 1; : : : ; tk , Ek;j
0 J1 .
Finally, the blocks with index 0, associated to the eigenvalue 0, are
C0 = diag(C0e ; C0+ ; C0, ); D0 = diag(D0e ; D0+ ; D0, ); F0 = diag(F0e ; F0+ ; F0, );
with substructures
C0e = D0e = diag(N2,x1 ; : : : ; N2,x ); F0e = , diag(N2+x1 ; : : : ; N2+x );
= 12 k;j

c

c

ELA
86

Structured Jordan Canonical Forms
#
"
#!
"
p2
p2
,
,
N
N
e
e
g
g
+
g
g
1
1
2
2
p
; : : : ; p2 T
;
C0 = diag
0
, 22 eTg1 0
, 2 eg
 N + 
 N + 
g
g
+
+
,
,
1
p
;
D0 = diag(Ng1 ; : : : ; Ng ); F0 = , diag
2 eT ; : : : ; p2 eT
g
2 1
2 g
!
p
p
2
2
+
+
,
,
,
,
C0 = diag(Nh1 ; : : : ; Nh ); F0 = diag [ 2 eh1 ; ,Nh1 ]; : : : ; [ 2 eh ; ,Nh ] ;
"
"
p2 T #!
p2 T #
0
0
e
,
2 eh
2 h1 ; : : : ;
p
p
:
D0 = diag
, 22 eh Nh,
, 22 eh1 Nh,1
Each nonzero real eigenvalue k (,k ) has lk Jordan blocks of sizes fk;1 ; : : : ; fk;l
and each nonreal eigenvalue k (,k ; k ; ,k ) that is not on the imaginary axis has
sk Jordan blocks with sizes pk;1 ; : : : ; pk;s .
For each nonzero purely imaginary eigenvalue k (,k ) we have
a) tk even sized Jordan blocks of sizes 2qk;1 ; : : : ; 2qk;t with the corresponding structure inertia indices i(,1)q 1 +1 k;1 ; : : : ; i(,1)q +1 k;t for k and i(,1)q 1 k;1 ; : : :,
i(,1)q k;t for ,k ;
b) wk odd sized Jordan blocks of sizes 2uk;1 + 1; : : : ; 2uk;w + 1 corresponding to the
structure inertia indices (,1)u 1 +1 ; : : : ; (,1)u +1 ;
c) zk odd sized Jordan blocks of sizes 2vk;1 + 1; : : : ; 2vk;z + 1 corresponding to the
structure indices (,1)v 1 ; : : : ; (,1)v .
The zero eigenvalue has 2c even sized Jordan blocks with sizes of 2x1 ; 2x1 , : : :,
2xc ; 2xc with corresponding structure inertia indices i; ,i; : : :; i; ,i , and a+b odd sized
Jordan blocks, where a of them have sizes 2g1 + 1; : : : ; 2ga + 1 with the corresponding structure inertia indices (,1)g1 +1 ; : : : ; (,1)g +1 and b of them have sizes 2h1 +
1; : : : ; 2hb + 1 with the corresponding structure inertia indices (,1)h1 ; : : : ; (,1)h .
a

a

a

a

a

a

b

b

b

b

b

b

k

k

k

k;tk

k;

k;tk

k;

k

k

k

k;wk

k;

k

k;

k;zk

a

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 U^c such that
 +



R
0
0
W
c
T
c
^
^
^
^
Uc p;q Uc = W T 0 ; C Uc = Uc 0 R, :
c
c
P
P
P
P
t

Let Uc := U^c diag(Im ; Wc,1 )m , where m := k=1 j=1 fk;j + k=1 sj=1 2pk;j . By
using Proposition 2.2 we can verify that
k

k

UcT p;q Uc = m;m;





CUc = Uc TRTc RTcc ;
c

where Rc ,Tc are as asserted.
For nonzero purely imaginary eigenvalues 1 ; : : :  we rst consider a single Jordan block Nr (Im J1 ). By Proposition 2.3 there exists a real matrix U^ such that
^ r (Im J1 );
U^ T p;q U^ = Pr
; C U^ = UN
where  = I2 if r is odd and  = (Im )J1 is r even, and  is the structure inertia
index corresponding to Nr (). If r is even, then we set U := U^ diag(Ir ; ((Im )P 2

J1 ),1 ) 2 and obtain

 ,

+ (02 ) + Er 
N
((Im

)
J
)
+
E
,
N
I
0
1
r
r
T
s
s
U p;q U = 0 ,I ; C U = U
,Ns+ (02 ) , Er Ns, ((Im )J1 ) , Er ;
r
r

r

ELA
87

Volker Mehrmann and Hongguo Xu




where Er = 21  00 J0
1
U

we have

:= U^ diag(Ir+1 ; (P r,2 1
I2 ),1 )(^ r,2 1
I2 )
2

U T p;q U

and  = (,1) 2r i. If r is odd, then we set

=4

3

Ir,1
2

6
6
C U = U 666
4

I2

,Ir,1

N,
r,1 ((Im  )J1 )
2
p
0 , 22 I2
,N +r,2 1 (02 )

5;

p20
2 I2

,N +r,2 1 (02 )
p
0 , 22 I2
N,
r,1 ((Im  )J1 )

(Im )J1
p0

, 22 I2

3
7
7
7
7;
7
5

2

2  . Based on these properties we can construct a real matrix Ug
where  = (,1) s+1
such that
 ^+ ^ 
UgT p;q Ug = n1 ;n2 ; CUg = Ug RT^gT RT^ ,g ;
g
g

where
R^ g+ = diag(C1 ; : : : ; C ); R^ g, = diag(D1 ; : : : ; D ); T^g = diag(F1 ; : : : ; F );
and Ck ; Dk ; Fk (k = 1; : : : ;  ) are in the asserted forms and
n1 = 2

n2 = 2


X
k=1

(wk +


X
k=1

(zk +

tk
X
j =1
tk
X
j =1

qk;j +

qk;j +

wk
X
j =1
wk
X
j =1

uk;j +

uk;j +

zk
X
j =1
zk
X
j =1

vk;j );

vk;j ):

For the eigenvalue zero we have distinguished between even and odd sized Jordan
blocks. For odd sized Jordan blocks as Nr there exists a real matrix U^ such that
^ r:
U^ T p;q U^ = Pr ; C U^ = UN
As in the purely imaginary case in the proof of Theorem 4.3 we then can generate a
real matrix U from U^ such that
2

U T p;q U

I r,2 1

=4 0
0
r+1

0

0
0

3

2

N,
r,1

6 p 2
5 ; C U = U 6 , 22 eTr,1

4
2
0 ,I r,2 1
,N +r,1
2

p2
2 e r,2 1
p

0

, 22 e r,2 1

,pN +r,2 1

, 22 eTr,2 1
N,
r ,1

3

7
7;
5

2

with  = (,1) 2 . By Proposition 2.3, even sized Jordan blocks Nr must appear in
pairs. More precisely for each pair of Nr ; Nr there is real matrix U^ such that




Nr 0
^U T p;q U^ = 0T Pr ;
^
^
C U = U 0 Nr :
Pr 0

ELA
88

Structured Jordan Canonical Forms

Hence with U := U^ diag(I

; Pr,1 )r we have


Ir
0 ; CU
U T p;q U =
0 ,Ir
r



,
+
= U ,NN + ,NN,
r

r

r


:

r

Based on these facts for the eigenvalue zero there exists a real matrix U such that
z

U  U =
T
z

p;q

z

n0 ;n0
2

1

CU = U

;

z



z

C0
F0T



F0
D0

;

Pc

Pa

where
C0 , F0 , D0 are in the asserted forms, and n01 = a +
=1 2x + =1 g +
P
02 = b + P 2x + P g + P h .
h
,
n
=1
=1
=1
=1
Finally by combining all these cases we can generate a real  -orthogonal matrix
U from U , U , U which satises (4.3).
We have seen that the results for  -Hermitian and  -skew Hermitian matrices 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  -unitary matrices.
5.  -unitary matrices. In the previous two sections we have studied structured Jordan canonical forms for  -Hermitian and  -skew Hermitian matrices.
Each class has an algebra structure, the  -Hermitian matrices form a Jordan algebra and the  -skew Hermitian matrices a Lie algebra. The Lie group associated
with these two algebras is the class of  -unitary matrices. In order to derive structured canonical forms for this group analogous to the results for the algebras, we can
make use of the Cayley transformation.
Lemma 5.1. If A is  -unitary and 1 62 (A) then the Cayley transformation
of B
b

c

k

k

a

k

k

b

k

k

k

k

k

k

k

k

p;q

c

g

z

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

p;q

B = (A) = (A + I )(A , I ),1
is  -skew Hermitian. Conversely, if A is  -skew Hermitian then B as in (5.1)
(5.1)

p;q

p;q

is  -unitary.
Proof. We only prove the result for the case that A is  -unitary. The other
direction follows form the fact that ((A)) = A.
Since A is  -unitary,  A = A,  . By this relation
p;q

p;q

p;q

H

p;q

p;q

 B =  (A + I )(A , I ),1 = (A, + I ) (A , I ),1
= (A, + I )(A, , I ),1  = (I + A )(A, )(A )(I , A ),1 
= (A + I ) (I , A),  = ,B  = ,( B) :
p;q

H

p;q

H

H

H

p;q
H

p;q

H

H

p;q

p;q

H

p;q

H

H

p;q

H

Therefore B is  -skew Hermitian.
Using the Cayley transformation  the canonical forms of  -unitary matrices
(if 1 is not an eigenvalue) can be easily obtained from the canonical form of the
corresponding  -skew Hermitian matrix discussed in Section 4. However, if we
Cayle