Relative Perturbation Bounds for Positive Polar Factors of Graded Matrices

Let B be an m × n (m ≥ n) complex (or real) matrix. It is known that there is a unique polar decomposition B = QH, where Q∗Q = I, the n× n identity matrix, and H is positive definite, provided that B has full column rank. If B is perturbed to B̃, 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 − H̃‖F ≤ √ 2‖B − B̃‖F in the Frobenius norm, due to [F. Kittaneh, Comm. Math. Phys., 104 (1986), pp. 307–310], where Q̃ and H̃ are the corresponding polar factors of B̃. 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 − H̃‖F means little, if anything, to the accuracy of H̃’s smaller entries. This paper proposes a new way of measuring the errors in the H factor via bounding the scaled difference (H̃ − H)S−1, as well as accurately computing the factor when S is diagonal. Numerical examples are presented. The results are also extended to the matrix square root of a graded positive definite matrix.