Bounds for the Rayleigh Quotient and the Spectrum of Self-Adjoint Operators


The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If x is an eigenvector of a self-adjoint bounded operator A in a Hilbert space, then the RQ of the vector x, denoted by ρ(x), is an exact eigenvalue of A. In this case, the absolute change of the RQ |ρ(x) - ρ(y)| becomes the absolute error for an eigenvalue ρ(x) of A approximated by the RQ ρ(y) on a given vector y. There are three traditional kinds of bounds for eigenvalue errors: a priori bounds via the angle between vectors x and y; a posteriori bounds via the norm of the residual Ay -ρ(y)y of vector y; mixed type bounds using both the angle and the norm of the residual. We propose a unifying approach to prove known bounds of the spectrum, analyze their sharpness, and derive new sharper bounds. The proof approach is based on novel RQ vector perturbation identities.


Computer Science

Keywords and Phrases

Angles; Bounded operators; Eigen-value; Eigenvalue errors; Perturbation; Rayleigh quotients; Self adjoint operator; Vector perturbation; Error analysis; Vectors; Eigenvalues and eigenfunctions; Eigenvalue

International Standard Serial Number (ISSN)

0895-4798; 1095-7162

Document Type

Article - Journal

Document Version


File Type





© 2013 Society for Industrial and Applied Mathematics, All rights reserved.

Publication Date

01 Jan 2013