PERTURBATION ANALYSIS OF THE EIGENVECTOR MATRIX AND SINGULAR VECTOR MATRICES

Abstract

Let $A$ be an $n \times n$ Hermitian matrix and $A = U \Lambda U^H$ be its spectral decomposition, where $U$ is a unitary matrix of order $n$ and $\Lambda$ is a diagonal matrix. In this note we present the perturbation bound and condition number of the eigenvector matrix $U$ of $A$ with distinct eigenvalues. A perturbation bound of singular vector matrices is also given for a real $n \times n$ or $(n+1) \times n$ matrix. The results are illustrated by numerical examples.