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

AbstractLet 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∑j=1nμπ(j)−λj2⩽n(1+n−p)max∥Q−1(B−A)Q∥F,∥Q−1(B−A)Q∥Fmand for j=1,…,n,μπ(j)−λj⩽n(1+n−p)maxn∥Q−1(B−A)Q∥2,n∥Q−1(B−A)Q∥2m,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