A Practical Upper Bound for Departure from Normality

A simple formula is presented for computing an upper bound for departure from normality in the Frobenius norm that is cheaper to compute than the one derived by Henrici and sharp for Hermitian matrices, skew-HermitianMatrices and, in general, any matrix with eigenvalues that are horizontally or vertically aligned in the complex plane.

Abstract

The departure from normality of a matrix is a real scalar that is impractical to compute if the matrix is large and its eigenvalues are unknown. A simple formula is presented for computing an upper bound for departure from normality in the Frobenius norm. This new upper bound is cheaper to compute than the one derived by Henrici [Numer. Math., 4 (1962), pp. 24–40]. Moreover, the new bound is sharp for Hermitian matrices, skew-Hermitian matrices and, in general, any matrix with eigenvalues that are horizontally or vertically aligned in the complex plane. In terms of applications, the new bound can be used in computing bounds for the spectral norm of matrix functions or bounds for the sensitivity of eigenvalues to matrix perturbations.