A generalization of Saad's theorem on Rayleigh–Ritz approximations

Abstract

Let (λ,x) be an eigenpair of the Hermitian matrix A of order n and let (μ,u) be a Ritz pair from a subspace K of C 2 . Y. Saad (Numerical Methods for Large Eigenvalue Problems: Theory and Algorithm, Wiley, New York, 1992) has given a simple inequality bounding sin ∠(x,u) in terms of sin ∠(x, K) . In this paper, we show that this inequality can be extended to an equally simple inequality for eigenspaces of non-Hermitian matrices.