Residual Bounds on Approximate Solutions for the Unitary Eigenproblem
Jiguang Sun · SIAM Journal on Matrix Analysis and Applications · 1996
Let A be an $n \times n$ unitary matrix, and let the columns of an $n \times l\,( l < n )$ matrix $\tilde X_1 $ form an orthonormal basis for an approximate eigenspace $\tilde{\mathcal{X}}_1 $ of A. Then there are two problems: How near is $\tilde{\mathcal{X}}_1 $ to an eigenspace of A? How can we make use of the $l \times l$ matrix $\tilde X_1^H A\tilde X_1 $ to get l approximate eigenvalues of A? This paper gives solutions to these problems. In particular, this paper reveals such a fact: One can use the eigenvalues of the unitary polar factor of $\tilde X_1^H A\tilde X_1 $ (or the eigenvalues of the matrix $\tilde X_1^H A\tilde X_1 $) as l approximate eigenvalues of A, and the precision of the eigenvalues of the unitary polar factor of $\tilde X_1^H A\tilde X_1 $ (or the eigenvalues of $\tilde X_1^H A\tilde X_1 $) as l approximate eigenvalues of A is higher than that of $\tilde{\mathcal{X}}_1 $ as an approximate eigenspace of A.