2. PRELIMINARIES 11

We conclude this section with the following two lemmas which allow us to

express the mvf G(z) defined by (1.4) in a certain canonical form which will simplify

some later computations.

LEMMA 2.10. Let 7\ and T2 be two invertible n x n matrices and let

M = TXMT2, N = TxNT2l P = T-*PT~\ and C = CT2. (2.9)

Then the problems aBIP(M, TV, P, C) and aBIPfM, TV, P, C) are equivalent to

aBIP(M, TV, P, 6) and aBIP^M, TV, P , C), respectively.

PROOF.

It suffices to note that

M*PM - TV*PTV = &JC,

and that

CG(zY

%x ) (M-zN)-1

C

l r p - l

(Ccl\{M- zN)-1^1

= CGizy'Ti

which implies that

P* = CG(()-\ ^ ( C ) -

1

] ^ =

[CGicy1^1, CG(C)-1T-1]S

= Tf* [CG(C)-1, CG(Q-l}sT^1

= r1-*p5T1-1.

D

The next lemma (for the proof see [15, Lemma 2.3]) is a slight variation of the

canonical representation for regular pencils (see e.g., [28, p. 28, Theorem 3]).

LEMMA

2.11. Let M and T V satisfy (1.5). Then there exist invertible matrices

Ti and T2 from

Cnxn

and matrices Ax G

CklXkl,

A2 e

Cfc2X/C2

and A3 e

Cfc3X/C3

with

spec Ax ( J spec A2 cB and spec A3 cT (2.10)

such that

' Ikl 0 0 \ / A1 0 0

TXMT2 = I 0 A2 0 \ and T±NT2 = 0 Ik2 0

0 0 0 0 A3

The following remark is an immediate consequence of the last two lemmas.

REMARK

2.12. Let condition (1.5) be in force. Then, without loss of generality,

the matrices M and T V from the data set (1.2) of the aBIP can be assumed to be

of the form

M =

(I*

0

V 0

0

A2

0

0

0

IK

and T V

A± 0 0

0 ik2 0

0 0 A3

(2.11)

where the Aj are matrices satisfying (2.10) and can be presumed to be in Jordan

form.