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
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.
Previous Page Next Page