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

AbstractThis 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−1AS is of first and S−1ZS 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