×

New perturbation bounds for the spectrum of a normal matrix. (English) Zbl 1370.15018

Summary: Let \(A \in \mathbb{C}^{n \times n}\) and \(\widetilde{A} \in \mathbb{C}^{n \times n}\) be two normal matrices with spectra \(\{\lambda_i \}_{i = 1}^n\) and \(\{\widetilde{\lambda}_i \}_{i = 1}^n\), respectively. The celebrated Hoffman-Wielandt theorem states that there exists a permutation \(\pi\) of \(\{1, \dots, n \}\) such that \((\sum_{i = 1}^n | \widetilde{\lambda}_{\pi(i)} - \lambda_i |^2)^{\frac{1}{2}}\) is no larger than the Frobenius norm of \(\widetilde{A} - A\). However, if either \(A\) or \(\widetilde{A}\) is non-normal, this result does not hold in general. In this paper, we present several novel upper bounds for \((\sum_{i = 1}^n | \widetilde{\lambda}_{\pi(i)} - \lambda_i |^2)^{\frac{1}{2}}\), provided that \(A\) is normal and \(\widetilde{A}\) is arbitrary. Some of these estimates involving the “departure from normality” of \(\widetilde{A}\) have generalized the Hoffman-Wielandt theorem. Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix.

MSC:

15A42 Inequalities involving eigenvalues and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Eisenstat, S. C.; Ipsen, I. C.F., Three absolute perturbation bounds for matrix eigenvalues imply relative bounds, SIAM J. Matrix Anal. Appl., 20, 149-158 (1998) · Zbl 0927.15009
[2] Henrici, P., Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math., 4, 24-40 (1962) · Zbl 0102.01502
[3] Hoffman, A. J.; Wielandt, H. W., The variation of the spectrum of a normal matrix, Duke Math. J., 20, 37-39 (1953) · Zbl 0051.00903
[4] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0729.15001
[5] Ipsen, I. C.F., Relative perturbation results for matrix eigenvalues and singular values, Acta Numer., 7, 151-201 (1998) · Zbl 0916.15008
[6] Ipsen, I. C.F., A note on unifying absolute and relative perturbation bounds, Linear Algebra Appl., 358, 239-253 (2003) · Zbl 1028.15016
[7] Kahan, W. M., Spectra of nearly Hermitian matrices, Proc. Amer. Math. Soc., 48, 11-17 (1975) · Zbl 0322.15022
[8] Li, R.-C., Relative perturbation theory: I. Eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl., 19, 956-982 (1998) · Zbl 0917.15009
[9] Li, W.; Sun, W., The perturbation bounds for eigenvalues of normal matrices, Numer. Linear Algebra Appl., 12, 89-94 (2005) · Zbl 1164.15329
[10] Li, W.; Vong, S.-W., On the variation of the spectrum of a Hermitian matrix, Appl. Math. Lett., 65, 70-76 (2017) · Zbl 1354.15014
[11] Song, Y., A note on the variation of the spectrum of an arbitrary matrix, Linear Algebra Appl., 342, 41-46 (2002) · Zbl 0995.15014
[12] Sun, J.-G., On the Wielandt-Hoffman theorem, Math. Numer. Sin., 2, 208-212 (1983) · Zbl 0522.15009
[13] Sun, J.-G., On the perturbation of the eigenvalues of a normal matrix, Math. Numer. Sin., 3, 334-336 (1984) · Zbl 0554.15009
[14] Sun, J.-G., On the variation of the spectrum of a normal matrix, Linear Algebra Appl., 246, 215-223 (1996) · Zbl 0867.15012
[15] Zhang, Z., On the perturbation of the eigenvalues of a non-defective matrix, Math. Numer. Sin., 6, 106-108 (1986) · Zbl 0611.15014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.