Pairs of matrices with a non-zero commutator

Abstract

1. This note takes its origin in a remark by Brauer ( 1 ) and Perfect ( 5 ): Let A be a square complex matrix of order n whose characteristic roots are α 1 ,…, α n . If X 1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X 1 k are α 1 + KX 1 , α 2 , …, α n . Recently, Goddard ( 2 ) extended this result as follows: If x 1 ; …, x r are linearly independent characteristic column vectors associated with the characteristic roots α 1 , …, α r of the matrix A , whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK , where K is an arbitrary r × n matrix, X = (x 1 , …, x r ) and Λ = diag ( α 1 , …, α r ). We shall prove some theorems of which these and other well-known results are special cases.