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 (x), is an exact eigenvalue of A. In this case, the absolute change of the RQ j(x) (y)j becomes the absolute error for an eigenvalue (x) of A approximated by the RQ (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 (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. SIAM Journal on Matrix Analysis and Applications This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c © Mitsubishi Electric Research Laboratories, Inc., 2013 201 Broadway, Cambridge, Massachusetts 02139 BOUNDS FOR THE RAYLEIGH QUOTIENT AND THE SPECTRUM OF SELF-ADJOINT OPERATORS ∗ PEIZHEN ZHU†‡, MERICO E. ARGENTATI†‡ AND ANDREW V. KNYAZEV†‡§¶ Abstract. 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. 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.