Bounds for the Relative and Absolute Spectral Variations of Matrices

Let A and A~ be n×n-matrices whose eigenvalues enumerated with their multiplicities are λ_k and λ~_j(j,k=1,..,n), respectively. In terms of the determinant of A and Frobenius norm of A~ we derive a bound for the relative spectral variation max_jmin_k|λ~_jλ_k-1| of A~ with respect to A, provided A is invertible. In addition, in terms of the Frobenius norm of A~, we obtain a new bound for the absolute spectral variation max_jmin_k|λ~_j-λ_k|. In appropriate situations our results are considerably sharper than the well-known bounds.