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

Ji-Guang Sun

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.

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@Article{JCM-9-247, author = {Sun , Ji-Guang}, 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 AU - Sun , Ji-Guang 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.

Sun , Ji-Guang. (1991). Rayleigh Quotient and Residual of a Definite Pair. Journal of Computational Mathematics. 9 (3). 247-255. doi:
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