Simultaneous reduction to triangular and companion forms of pairs of matrices: the case rank(I−-AZ)=1

Abstract This paper is concerned with simultaneous reduction to triangular and companion forms of pairs of matrices A , Z with rank( I − AZ )=1. Special attention is paid to the case where A is a first and Z is a third companion matrix. Two types of simultaneous triangularization problems are considered: (1) the matrix A is to be transformed to upper triangular and Z to lower triangular form, (2) both A and Z are to be transformed to the same (upper) triangular form. The results on companions are made coordinate free by characterizing the pairs A , Z for which there exists an invertible matrix S such that S −1 AS is of first and S −1 ZS is of third companion type. One of the main theorems reads as follows: If rank( I − AZ )=1 and αζ ≠1 for every eigenvalue α of A and every eigenvalue ζ of Z , then A and Z admit simultaneous reduction to complementary triangular forms.