Accuracy of Computed Eigenvectors Via Optimizing a Rayleigh Quotient

This paper establishes converses to the well-known result: for any vector such that the sine of the angle sin(u, )=O(), we have ,where is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and ^* denotes complex conjugate transpose. It shows that if ( ) is close to A's largest eigenvalue, then is close to the corresponding eigenvector with an error proportional to the square root of the error in ( ) as an approximation to the eigenvalue and inverse proportional to the square root of the gap between A's first two largest eigenvalues. A subspace version of such an converse is also established. 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.