Perturbation analysis for best approximation and the polar factor by subunitary matrices

Abstract

This paper is a continuation and improvement over the results of Laszkiewicz and Zietak [BIT, 2006, 46: 345–366], studying perturbation analysis for polar decomposition. Some basic properties of best approximation subunitary matrices are investigated in detail. The perturbation bounds of the polar factor are also derived.

Keywords

Best approximation by subunitary matrices / minimal rank approximation / polar decomposition

References

[1] Araki H., Yamagami S. An inequality for the Hilbert-Schmidt norm. Comm Math Phys, 1981, 81: 89-98.

[2] Autonne L. Sur les groupes linéaires, réel et orthogonaux. Bull Soc Math France, 1902, 30: 121-134.

[3] Barrlund A. Perturbation bounds on the polar decomposition. BIT, 1989, 30: 101-113.

[4] Chatelin F., Gratton S. On the condition numbers associated with the polar factorization of a matrix. Numerical Linear Algebra with Appl, 2000, 7: 337-354.

[5] Chen C., Sun J. G. Perturbation bounds for the polar factors. J Comput Math, 1989, 7: 397-401.

[6] Chen X., Li W. Relative perturbation bounds for the subunitary polar factor under unitarily invariant norms. Adv Math (in Chinese), 2006, 35: 178-184.

[7] Chen X., Li W., Sun W. Some new perturbation bounds for the generalized polar decomposition. BIT, 2004, 44: 237-244.

[8] Cucker F., Diao H., Wei Y. Smoothed analysis of some condition numbers. Numer Linear Algebra Appl, 2006, 13: 71-84.

[9] Golub G. H., Van Loan C. F. Matrix Computations, 1996 3rd Ed Baltimore: The Johns Hopkins University Press.

[10] Higham N. J. Computing the polar decomposition—with applications. SIAM J Sci Stat Comput, 1986, 7: 1160-1174.

[11] Higham N. J. The matrix sign decomposition and its relation to the polar decomposition. Linear Algebra Appl, 1994, 212/213: 3-20.

[12] Higham N. J., Mackey D. S., Mackey N. Computing the polar decomposition and the matrix sign decomposition in matrix groups. SIAM J Matrix Anal Appl, 2004, 25: 1178-1192.

[13] Higham N. J., Schreiber R. S. Fast polar decomposition of an arbitrary matrix. SIAM J Sci Stat Comput, 1990, 11: 648-655.

[14] Kittaneh F. Inequalities for the Schatten p-norm, III. Comm Math Phys, 1986, 104: 307-310.

[15] Laszkiewicz B., Zietak K. Approximation of matrices and a family of Gander methods for polar decomposition. BIT, 2006, 46: 345-366.

[16] Li R. C. A perturbation bound for the generalized polar decomposition. BIT, 1993, 33: 304-308.

[17] Li R. C. New perturbation bounds for the unitary polar factor. SIAM J Matrix Anal Appl, 1995, 16: 327-332.

[18] Li R. C. Relative perturbation bounds for the unitary polar factor. BIT, 1997, 37: 67-75.

[19] Li R. C. Relative perturbation bounds for positive polar factors of graded matrices. SIAM J Matrix Anal Appl, 2005, 27: 424-433.

[20] Li W. Some new perturbation bounds for subunitary polar factors. Acta Math Sin (Engl Ser), 2005, 21: 1515-1520.

[21] Li W., Sun W. Perturbation bounds of unitary and subunitary polar factors. SIAM J Matrix Anal Appl, 2002, 23: 1183-1193.

[22] Li W., Sun W. New perturbation bounds for unitary polar factors. SIAM J Matrix Anal Appl, 2003, 25: 362-372.

[23] Li W., Sun W. Some remarks on the perturbation of polar decomposition for rectangular matrices. Numerical Linear Algebra Appl, 2006, 13: 327-338.

[24] Maher P. Some matrix approximation problems arising from quantum chemistry. Proc Indian Natl Sci Acad, 1998, 64: 715-723.

[25] Mathias R. Perturbation bounds for the polar decomposition. SIAM J Matrix Anal Appl, 1993, 14: 588-597.

[26] Mirsky L. Symmetric gauge functions and unitarily invariant norm. Quart J Math Oxford, 1960, 11: 50-59.

[27] Sun J. G. Matrix Perturbation Analysis, 2001 2nd Ed Beijing: Science Press.

[28] Sun J. G., Chen C. H. Generalized polar decomposition. Math Numer Sinica, 1989, 11: 262-273.

[29] Wedin P. Perturbation theory for the pseudoinverse. BIT, 1973, 13: 217-232.

[30] Wei M. Perturbation theory for the Echart-Young-Mirsky theorem and the constrained total least squares problem. Linear Algebra Appl, 1998, 280: 267-287.