Accuracy of Computed Eigenvectorsvia Optimizing a Rayleigh
نویسنده
چکیده
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(2) where is an eigenvalue and u is the corresponding eigenvector of a Her-mitian 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. Abstract This note gives converses to the well-known result: for any vector e u such that
منابع مشابه
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 ...
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