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arXiv · 1308.1122

The minimization of matrix logarithms - on a fundamental property of the unitary polar factor

Johannes Lankeit·Patrizio Neff·Yuji Nakatsukasa
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Abstract

We show that the unitary factor U in the polar decomposition of a nonsingular matrix Z = U H is the minimizer for both ||Log(Q^* Z)|| and ||sym (Log(Q^*Z))|| over unitary Q, for any given invertible complex n-times-n matrix Z, for any unitarily invariant norm and any n. We prove that U is the unique matrix with this property. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.

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Johannes Lankeit, Patrizio Neff, Yuji Nakatsukasa. 2013-08-05. The minimization of matrix logarithms - on a fundamental property of the unitary polar factor. https://arxiv.org/abs/1308.1122

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