A note on the variation of the spectrum of an arbitrary matrix

Abstract

Let A and B be two n× n matrices with spectra λ( A)={ λ 1,…, λ n } and λ( B)={ μ 1,…, μ n }. Suppose that the nonsingular matrix Q satisfies Q −1 AQ=diag( J 1,…, J p ), where each submatrix J i, i=1,…,p , is a Jordan block. Then there exists a permutation π of {1,…, n} such that ∑ j=1 n μ π(j)−λ j 2 ⩽ n (1+ n−p ) max ∥Q −1(B−A)Q∥ F , ∥Q −1(B−A)Q∥ F m and for j=1,…, n, μ π(j)−λ j ⩽ n (1+ n−p ) max n ∥Q −1(B−A)Q∥ 2, n ∥Q −1(B−A)Q∥ 2 m , where m is the order of the largest Jordan block of A and ∥ ∥ F and ∥ ∥ 2 denote, respectively, the Frobenius norm and the spectral norm.