PAIRS OF MATRICES AND THEIR SIMULTANEOUS REDUCTION TO TRIANGULAR AND COMPANION FORMS IN THE CASE OF RANK(I — AZ) =1

This paper is concerned with simultaneous reduction to triangular and companion forms of pairs of matrices A, Z with rank( / — 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: ( ) 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(/ — AZ) =1 and saC 1 for every eigenvalue a of A and every eigenvalue of Z, then A and Z admit simultaneous reduction to complementary triangular forms