Pairs of matrices with a non-zero commutator

Extract

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 α₁,…, α_n. If X₁ is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X₁ k are α₁ + KX₁, α₂, …, αⁿ. Recently, Goddard (2) extended this result as follows: If x₁; …, x_r are linearly independent characteristic column vectors associated with the characteristic roots α₁, …, 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 = (x₁, …, x_r) and Λ = diag (α₁, …, α_r). We shall prove some theorems of which these and other well-known results are special cases.