Abstract
In this paper, we study the perturbation bounds for the polar decomposition A = QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ 2 r ‖ΔQ‖ 2 F ⩽ ‖ΔA‖ 2 F , 1/2‖ΔH‖ 2 F ⩽ ‖ΔA‖ 2 F and ‖ΔΣ‖ 2 F ⩽ ‖ΔA‖ 2 F , respectively, where Σ = diag(σ 1, σ 2, …, σ r , …, 0 ) is the singular value matrix of A and σ r denotes the smallest nonzero singular value. Here we present some new combined (asymptotic) perturbation bounds σ 2 r ‖ΔQ‖ 2 F +1/2‖ΔH‖ 2 F ⩽ ‖ΔA‖ 2 F and σ 2 r ‖ΔQ‖ 2 F +‖ΔΣ‖ 2 F ⩽ ‖ΔA‖ 2 F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Sun J G, Chen C H. Generalized polar decomposition. Math Numer Sinica, 11: 262–273 (1989)
Higham N J. Computing the polar decomposition-with applications. SIAM J Sci Statist Comput, 7: 1160–1174 (1986)
Barrlund A. Perturbation bounds on the polar decomposition. BIT, 30: 101–113 (1989)
Bhatia R. Matrix Analysis. New York: Springer, 1997
Bhatia R, Mukherjea K. Variation of the unitary part of a matrix. SIAM J Matrix Anal Appl, 15: 1007–1014 (1994)
Chatelin F, Gratton S. One the condition numbers associated with the polar factorization of a matrix. Numer Linear Algebra Appl, 7: 337–354 (2000)
Li R C. A perturbation bounds for the generalized polar decomposition. BIT, 33: 304–308 (1993)
Li R C. New Perturbation bounds for the unitary polar factor. SIAM J Matrix Anal Appl, 16: 327–332 (1995)
Li W, Sun W W. Perturbation bounds for unitary and subunitary polar factors. SIAM J Matrix Anal Appl, 23: 1183–1193 (2002)
Li W, Sun W W. New perturbation bounds for unitary polar factors. SIAM J Matrix Anal Appl, 25: 362–372 (2003)
Chen X S, Li W, Sun W W. Some new perturbation bounds for the generalized polar decomposition. BIT, 44: 237–244 (2004)
Mathias R. Perturbation bounds for the polar decomposition. SIAM J Matrix Anal Appl, 14: 588–593 (1993)
Horn R A, Johnson C R. Matrix Analysis. Cambridge: Cambridge University Press, 1985
Stewart G W, Sun J G, Matrix Perturbation Theory. Boston: Academic Press, 1990
Li W, Sun W W. Combined perturbation bounds: I. Eigensystems and singular value decomposition. SIAM J Matrix Anal Appl, 29: 643–655 (2007)
Chang X C, Paige C C, Stewart G W. Perturbation analysis for the QR decomposition. SIAM J Matrix Anal Appl, 18: 775–791 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the National Natural Science Foundation of China (Grant No. 10671077), the Natural Science Foundation of Guangdong Province (Grant Nos. 06025061, 031496) and the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102204)
Rights and permissions
About this article
Cite this article
Li, W., Sun, Ww. Combined perturbation bounds: II. Polar decompositions. SCI CHINA SER A 50, 1339–1346 (2007). https://doi.org/10.1007/s11425-007-0099-z
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11425-007-0099-z