From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Jordan decomposition
Date: 22 Dec 2000 22:45:43 -0500
Newsgroups: sci.math
Summary: [missing]

In article <3A43FC4F.35CC8EF4@penmaen.demon.co.uk>,
Dr. Uwe Prells <prells@penmaen.demon.co.uk> wrote:
:Hello and a merry Chistmas!
:
:Unfortunately research never stops.
:
:We know that for an arbitrary n-by-n matrix A the Jordan
:decomposition A=Z*J_A*Z^{-1} is one of the closest decompo to
:diagonalisation  if there are repeated defective eigenvalues of A.
:Is it true that for the general case A+ lambda*B there always
:exist matrices U,V (non-singular) such that
: U*(A+lambda*B)*V = J_A+lambda*J_b, 
:where J_A and J_B are the Jordan matrices
:of A,B resp. ?
:
 The key phrases are "matrix pencil", "Kronecker canonical form",
and for numerical purposes, "Generalized Schur Form".

 Generalized Schur Form gives two unitary matrices Q and Z such
that  T := Q'*A*Z   and   S := Q'*B*Z  are upper triangular. This
gives the pairs (thought of as coordinates of points in the projective
complex plane, except for degenerate situations)  t_kk and s_kk ; 

for s_kk not 0,   t_kk/s_kk is an eigenvalue from L(A,B),
for t_kk not 0,   s_kk/t_kk is an eigenvalue from L(B,A)

(casually we would say that the pair (A,B) has an 
"infinite eoigenvalue" if s_kk = 0 but t_kk is not 0).

Degenerate case: if for some k, t_kk=s_kk=0, then the whole
complex plane is the eigenvalue set of the pair (A,B).

If you want to go further, the simultaneous triangular form can be
transformed (by nonsingular matrices, not necessarily unitary)
to a Kronecker form which may, in degenerate cases, contain
non-square blocks of the pattern illustrated by

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

and their transposes.

And the situation is about the same, with some extra zero blocks,
even when A and B are non-square but still of the same size.

The book
 Matrix Computations, by Golub and Van Loan, talks about the non-
degenerate cases in section 7.7 (of my 3rd edition, 
Johns Hopkins 1996, ISBN 0-8018-5414-8).

The book also has references about the most general case (John
Wilkinson and others).

Happy holidays, ZVK(Slavek).