On condensed forms for partially commuting matrices

Two complex n x n matrices A and B are said to be partially commuting if A and B have a common eigenvector. We propose a condensed form for such matri ces that can be obtained from A and B by a finite rational computation. The condensed form is a pair of block triangular matrices, with the sizes of t he blocks being uniquely defined by the original matrices, We then show how to obtain additional zeros inside the diagonal blocks of a condensed form by using the generalized Lanczos procedure as given by Elsner and Ikramov. This procedure can also be considered as a finite rational process. We poin t out several applications of the constructions above. It turns out that fo r Laffey pairs of matrices, i.e., for matrices (A, B) such that rank[A, B] = 1, the condensed form is a pair of 2 x 2 block triangular matrices. Using this fact, we show an economical way to find a spanning set for the matrix algebra,generated by Laffey matrices A and B. Another application concerns so-called k-self-adjoint matrices. We examine such matrices in the unitary space as well as in a Krein space of defect 1. As an Appendix, we give a n ew description of the Shemesh subspace of matrices A and B. This is the max imal common invariant subspace of A and B, on which these matrices commute. (C) 2000 Elsevier Science Inc. All rights reserved.