Complementary Triangular Forms of Upper Triangular Toeplitz Matrices

The problem considered is the following. Given two upper triangular Toeplitz 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? The motivation for considering simultaneous reduction to complementary triangular forms of pairs of matrices comes from systems theory. For upper triangular Toeplitz matrices, a complete answer is given. The argument involves a detailed analysis of a certain directed graph associated with A and Z. Along the way information is obtained about the structure of the similarity S . The results actually hold for a class of matrices strictly larger than that consisting of the upper triangular Toeplitz matrices. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.