On the condition numbers associated with the polar factorization of a matrix

We are interested in the calculation of explicit formulae for the condition numbers of the two factors of the polar decomposition of a full rank real or complex m × n matrix A, where m ≥ n. We use a unified presentation that enables us to compute such condition numbers in the Frobenius norm, in cases where A is a square or a rectangular matrix subjected to real or complex perturbations. We denote by σ1 (respectively σn) the largest (respectively smallest) singular value of A, and by K(A) = σ1/σn the generalized condition number of A. Our main results are that the absolute condition number of the Hermitian polar factor is √2(1 + K(A)2)1/2/(1 + K(A)) and that the absolute condition number of the unitary factor of a rectangular matrix is 1/σn. Copyright © 2000 John Wiley & Sons, Ltd.

[1]  Ren-Cang Li,et al.  A perturbation bound for the generalized polar decomposition , 1993 .

[2]  Ren-Cang Li,et al.  Relative perturbation bounds for the unitary polar factor , 1997 .

[3]  S. Gratton,et al.  Structured Backward Error and Condition Number for Linear Systems of the Type A* Ax = b , 2000 .

[4]  Tosio Kato.  Continuity of the Map S→|S| for Linear Operators , 1973 .

[5]  J. Rice.  A Theory of Condition , 1966 .

[6]  H. Araki,et al.  An inequality for Hilbert-Schmidt norm , 1981 .

[7]  N. Higham.  The matrix sign decomposition and its relation to the polar decomposition , 1994 .

[8]  Fuad Kittaneh,et al.  Inequalities for the Schattenp-norm. III , 1986 .

[9]  N. Higham.  Computing the polar decomposition with applications , 1986 .

[10]  Charles Kenney,et al.  Polar Decomposition and Matrix Sign Function Condition Estimates , 1991, SIAM J. Sci. Comput..

[11]  Françoise Chaitin-Chatelin,et al.  Lectures on finite precision computations , 1996, Software, environments, tools.

[12]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[13]  R. Mathias.  Perturbation Bounds for the Polar Decomposition , 1997 .

[14]  Anders Barrlund,et al.  Perturbation bounds on the polar decomposition , 1990 .

[15]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

[16]  R. Bhatia.  Matrix factorizations and their perturbations , 1994 .