Rayleigh Quotient and Residual of a Definite Pair

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.