A note on the variation of the spectrum of an arbitrary matrix Original Research Article

Let A and B be two n×n matrices with spectra λ(A)={λ1,…,λn} and λ(B)={μ1,…,μn}. Suppose that the nonsingular matrix Q satisfies Q−1AQ=diag(J1,…,Jp), where each submatrix Ji, i=1,…,p, is a Jordan block. Then there exists a permutation π of {1,…,n} such that image and for j=1,…,n, image where m is the order of the largest Jordan block of A and short parallel short parallelF and short parallel short parallel2 denote, respectively, the Frobenius norm and the spectral norm.