Bounds for the Relative and Absolute Spectral Variations of Matrices

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