A Bidiagonal Matrix Determines Its Hyperbolic SVD to Varied Relative Accuracy

Abstract

Let $T = L \Omega L^t$ be an invertible, unreduced, indefinite tridiagonal symmetric matrix with $\Omega$ a diagonal signature matrix. We provide error bounds on the (relative) change in an eigenvalue and the angular change in its eigenvector when the entries in L suffer small relative changes. Our results extend those of Demmel and Kahan for $\Omega = I$. The relative condition number for an eigenvalue exceeds by 1 its absolutecondition number as an eigenvalue of $\Omega L^t L$. The condition number of an eigenvector is a weighted sum of the relative separations of the eigenvalue from each of the others. A small example shows that very small eigenpairs can be robust even when the large eigenvalues are extremely sensitive. When L is well conditioned for inversion, then all eigenvalues are robust and the eigenvectors depend only on the relative separations.