NDLI: Relative perturbation bounds for the unitary polar factor

Content Provider	Springer Nature Link
Author	Li, Ren Cang
Copyright Year	1997
Abstract	LetB be anm×n (m≥n) complex (or real) matrix. It is known that there is a uniquepolar decomposition B=QH, whereQQ=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.
Ending Page	75
Page Count	9
Starting Page	67
File Format	PDF
ISSN	00063835
e-ISSN	15729125
Journal	BIT Numerical Mathematics
Issue Number	1
Volume Number	37
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	1997-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Eigenvalues, singular values, and eigenvectors relative perturbation theory scaling Computational Mathematics and Numerical Analysis Factorization of matrices Numeric Computing Mathematics polar decomposition Matrix norms, conditioning, scaling
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

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