Bounds for the relative and absolute spectral variations of matrices (Q6561362)

Matrix eigenvalue perturbation bounds have been extensively studied for decades. Let \(A\) and \(\tilde{A}\) be \(n\times n\)-matrices whose eigenvalues, enumerated with their multiplicities, are \(\lambda_k\) and \(\tilde{\lambda}_j\) \((j, k = 1, \ldots, n)\), respectively. The author presents bounds for the relative spectral variation of \(\tilde{A}\) with respect to \(A\), defined as \(\max\limits_j\min\limits_k|\frac{\tilde{\lambda}_j}{\lambda_k}-1|\), and the absolute (or additive) spectral variation, defined as \(\max\limits_j\min\limits_k|\tilde{\lambda}_j-\lambda_k|\). The bound is in terms of determinants or Frobenius norms of the underlying matrices. It is stated that in certain situations the new results are considerably sharper than well-known bounds in the literature.