Volume 9, Issue 3
Rayleigh Quotient and Residual of a Definite Pair
DOI:

J. Comp. Math., 9 (1991), pp. 247-255

Published online: 1991-09

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• Abstract

Let {A,B} be a definite matrix pair of order n, and let Z be an l-dimensional subspace of $C^n$. In this paper we introduce the Rayleigh quotient matrix pair {$H_1,K_1$} and residual matrix pair {$R_A,R_B$} of {A,B} with respect to Z, and used the norm of {$R_A,R_B$} to bound the difference between the eigenvalues of {$H_1,K_1$} and that of {A,B}, and to bound the difference between Z and an l-dimensional eigenspace of {A,B}. The corresponding classical theorems on the Hermitian matrices can be derived from the results of this paper.

• Keywords

@Article{JCM-9-247, author = {}, title = {Rayleigh Quotient and Residual of a Definite Pair}, journal = {Journal of Computational Mathematics}, year = {1991}, volume = {9}, number = {3}, pages = {247--255}, abstract = { Let {A,B} be a definite matrix pair of order n, and let Z be an l-dimensional subspace of $C^n$. In this paper we introduce the Rayleigh quotient matrix pair {$H_1,K_1$} and residual matrix pair {$R_A,R_B$} of {A,B} with respect to Z, and used the norm of {$R_A,R_B$} to bound the difference between the eigenvalues of {$H_1,K_1$} and that of {A,B}, and to bound the difference between Z and an l-dimensional eigenspace of {A,B}. The corresponding classical theorems on the Hermitian matrices can be derived from the results of this paper. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9398.html} }
TY - JOUR T1 - Rayleigh Quotient and Residual of a Definite Pair JO - Journal of Computational Mathematics VL - 3 SP - 247 EP - 255 PY - 1991 DA - 1991/09 SN - 9 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9398.html KW - AB - Let {A,B} be a definite matrix pair of order n, and let Z be an l-dimensional subspace of $C^n$. In this paper we introduce the Rayleigh quotient matrix pair {$H_1,K_1$} and residual matrix pair {$R_A,R_B$} of {A,B} with respect to Z, and used the norm of {$R_A,R_B$} to bound the difference between the eigenvalues of {$H_1,K_1$} and that of {A,B}, and to bound the difference between Z and an l-dimensional eigenspace of {A,B}. The corresponding classical theorems on the Hermitian matrices can be derived from the results of this paper.