Relative Perturbation Bounds for Positive Polar Factors of Graded Matrices

Abstract

Let $B$ be an $m\times n$ ($m\ge n$) complex (or real) matrix. It is known that there is a unique {\em polar decomposition} $B=QH$, where $Q^*Q=I$, the $n\times n$ identity matrix, and $H$ is positive definite, provided that $B$ has full column rank. If $B$ is perturbed to $\wtd B$, how do the polar factors $Q$ and $H$ change? This question has been investigated quite extensively, but most work so far has been on how the perturbation changed the unitary polar factor $Q$, with very little on the positive polar factor $H$, except $\|H-\wtd H\|_{\F}\le\sqrt 2\|B-\wtd B\|_{\F}$ in the Frobenius norm, due to [F. Kittaneh, {\em Comm. Math. Phys.}, 104 (1986), pp. 307--310], where $\wtd Q$ and $\wtd H$ are the corresponding polar factors of $\wtd B$. While this inequality of Kittaneh shows that $H$ is always well behaved under perturbations, it does not tell much about smaller entries of $H$ in the case when $H$'s entries vary a great deal in magnitude. This paper is intended to fill the gap by addressing the variations of $H$ for a graded matrix $B=GS$, where $S$ is a scaling matrix and usually diagonal (but may not be). The elements of $S$ can vary wildly, while $G$ is well conditioned. In such cases, the magnitudes of $H$'s entries indeed often vary a lot, and thus any bound on $\|H-\wtd H\|_{\F}$ means little, if anything, to the accuracy of $\wtd H$'s smaller entries. This paper proposes a new way of measuring the errors in the $H$ factor via bounding the scaled difference $(\wtd H-H)S^{-1}$, as well as accurately computing the factor when $S$ {\em is\/} diagonal. Numerical examples are presented. The results are also extended to the matrix square root of a graded positive definite matrix.