A Generalization of Saad's Theorem on Rayleigh-Ritz Approximations

Abstract

Let $(\lambda,x)$ be an eigenpair of the Hermitian matrix $A$ of order $n$ and let $(\mu,u)$ be a Ritz pair from a subspace $\clk$ of $\comp^{2}$. Saad has given a simple inequality bounding $\sin\angle(x,u)$ in terms of $\sin\angle(x,\clk)$. In this note we show that this inequality can be extended to an equally simple inequality for eigenspaces of non-Hermitian matrices. (Also cross-referenced as UMIACS-TR-99-78)