Similarity invariants for pairs of upper triangular Toeplitz matrices

Abstract

The problem considered is that of simultaneous reduction to simple forms of pairs of upper triangular Toeplitz matrices. In this context, simple matrices are those that are equal to, or differ only slightly from, a power of the upper triangular nilpotent Jordan block. The problem is related to that of complementary triangularization of pairs of matrices and (therefore) has a background in systems theory. The main results are concerned with existence and uniqueness of the reduced form. Also a similarity invariant for pairs of upper triangular Toeplitz matrices is obtained. The invariant consists of a pair involving a complex number and an integer. The results can be generalized to a class of matrices strictly larger than that formed by the upper triangular Toeplitz matrices.