Rayleigh-Ritz Majorization Error Bounds of Mixed Type
Abstract
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with respect to vectors is bounded in terms of the norms of the residual vectors and the angle between vectors in [P. Zhu, M. E. Argentati, and A. V. Knyazev, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 244-256]. We substitute multidimensional subspaces for the vectors and derive new bounds of absolute changes of eigenvalues of the matrix RQ in terms of singular values of residual matrices and principal angles between subspaces, using majorization. We show how our results relate to bounds for eigenvalues after discarding off-diagonal blocks or additive perturbations.
Recommended Citation
P. Zhu and A. V. Knyazev, "Rayleigh-Ritz Majorization Error Bounds of Mixed Type," SIAM Journal on Matrix Analysis and Applications, vol. 38, no. 1, pp. 30 - 49, Society for Industrial and Applied Mathematics (SIAM), Jan 2017.
The definitive version is available at https://doi.org/10.1137/16M1058121
Department(s)
Computer Science
Keywords and Phrases
Eigenvalues and eigenfunctions; Error analysis; Matrix algebra; Angles; Eigen-value; Hermitians; Majorization; Rayleigh; Ritz; Subspaces; Vectors; Matrix
International Standard Serial Number (ISSN)
0895-4798; 1095-7162
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2017 Society for Industrial and Applied Mathematics (SIAM), All rights reserved.
Publication Date
01 Jan 2017
Comments
The work of the first author was supported by the National Science Foundation through grant DMS-1115734.