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.