Accuracy of computed eigenvectors via optimizing a Rayleigh quotient

This paper establishes converses to the well-known result: for any vector (u) over tilde such that the sine of the angle sin theta(u, (u) over tilde) = O(epsilon), we have rho((u) over tilde) (def)(=) (u) over tilde* A (u) over tilde/(u) over tilde*(u) over tilde = lambda + O((2)(epsilon)), where lambda is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and * denotes complex conjugate transpose. It shows that if rho((u) over tilde) is close to A's largest eigenvalue, then (u) over tilde is close to the corresponding eigenvector with an error proportional to the square root of the error in rho((u) over tilde) 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.