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Relative perturbation bounds for the unitary polar factor

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Abstract

LetB be anm×n (m≥n) complex (or real) matrix. It is known that there is a uniquepolar decomposition B=QH, whereQ*Q=I, then×n identity matrix, andH is positive definite, providedB has full column rank. Existing perturbation bounds suggest that in the worst case, for complex matrices the change inQ be proportional to the reciprocal ofB's least singular value, or the reciprocal of the sum ofB's least and second least singular values if matrices are real. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much mayQ change ifB is perturbed to\(\tilde B = D_1^* BD_2 \), whereD 1 andD 2 are nonsingular and close to the identity matrices of suitable dimensions? It is shown that for a such kind of perturbation, the change inQ is bounded only by the distances fromD 1 andD 2 to identity matrices and thus is independent ofB's singular values. Such perturbation is restrictive, but not unrealistic. We show how a frequently used scaling technique yields such a perturbation and thus scaling may result in better-conditioned polar decompositions.

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Authors and Affiliations

  1. Oak Ridge National Laboratory, Mathematical Science Section, Bldg 6012, P.O. Box 2008, 37831-6367, Oak Ridge, TN

    Ren-Cang Li

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  1. Ren-Cang Li
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This material is based in part upon work supported, during January, 1992–August, 1995, by Argonne National Laboratory under grant No. 20552402 and by the University of Tennessee through the Advanced Research Projects Agency under contract No. DAAL03-91-C-0047, by the National Science Foundation under grant No. ASC-9005933, and by the National Science Infrastructure grants No. CDA-8722788 and CDA-9401156, and supported, since August, 1995, by a Householder Fellowship in Scientific Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, Office of Energy Research, United States Department of Energy contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp.

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Li, RC. Relative perturbation bounds for the unitary polar factor. Bit Numer Math 37, 67–75 (1997). https://doi.org/10.1007/BF02510173

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  • DOI: https://doi.org/10.1007/BF02510173

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