Accuracy of Computed Eigenvectors via Optimizing a Rayleigh Quotient 1 Accuracy of Computed Eigenvectors via Optimizing a Rayleigh Quotient

This note gives converses to the well-known result: for any vector e u such that sin (u; e u) = O( ), we have e u Ae u e u e u = + O( ) where is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and \ " denotes complex conjugate transpose. It shows that if e u Ae u=e u e u is close to A's largest eigenvalue, then e u is close to the corresponding eigenvector with an error proportional to the square root of the error in e u Ae u=e u e u as an approximation to the eigenvalue and inverse proportional to the square root of the gap between A's rst two largest eigenvalues. We also have a subspace version of such an converse. Results as such may have interest in applications, such as eigenvector computations in Principal Component Analysis in image processing where eigenvectors may be computed by optimizing Rayleigh quotients with the Conjugate Gradient method. Let A be Hermitian with an eigenvalue and the corresponding eigenvector u. The following is well-known: For any vector e u such that sin (u; e u) = O( ), we have e u Ae u e u e u = + O( ) (1) (see, e.g., [4, 5]), where \ " denotes complex conjugate transpose. But I have not seen any converse to this statement appearing thus far. Such a converse is of interest to situations, e.g., eigenvector computations in Principal Component Analysis in image processing [3, 6], where eigenvectors may be computed by optimizing Rayleigh quotients with the Conjugate Gradient method. One may expect that if (1) holds then e u would be close to an eigenvector u of A. But this may not be true without additional assumptions. Here are two examples: Department of Mathematics, University of Kentucky, Lexington, KY 40506, email: [email protected]. This work was supported in part by the National Science Foundation (NSF) under Grant No. ACI-9721388 and a NSF/CAREER award under Grant No. CCR-9875201.