Complementary triangular forms of pairs of matrices, realizations with prescribed main matrices, and complete factorization of rational matrix functions

Abstract

The problem considered is the following. Given two square matrices A and Z, when does there exist an invertible matrix S such that S -1 AS is upper triangular and S -1 ZS is lower triangular? Several sufficient conditions for such an S to exist are presented. In some cases, the conditions are also necessary. Attention is paid to the order in which the eigenvalues of A and Z may appear on the diagonals of S -1 AS and S -1 ZS, respectively. There is an intimate connection with the notions of realization and minimal factorization from systems theory.