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Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25:362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A − Ã‖2 ≪ ‖A − Ã‖F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14:588–593) for both real and complex square matrices are generalized. Copyright © 2005 John Wiley & Sons, Ltd.