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The minimization of matrix logarithms: On a fundamental property of the unitary polar factor

Authors
  • Lankeit, Johannes
  • Neff, Patrizio
  • Nakatsukasa, Yuji1, 2, 3, 1, 2, 4, 5, 6
  • 1 Fakultät für Mathematik
  • 2 Universität Duisburg-Essen
  • 3 Lehrstuhl für Nichtlineare Analysis und Modellierung
  • 4 Department of Mathematical Informatics
  • 5 Graduate School of Information Science and Technology
  • 6 University of Tokyo
Type
Published Article
Journal
Linear Algebra and its Applications
Publisher
Elsevier
Publication Date
Jan 01, 2014
Accepted Date
Feb 04, 2014
Volume
449
Pages
28–42
Identifiers
DOI: 10.1137/130920137
Source
Elsevier
Keywords
License
Unknown

Abstract

We show that the unitary factor Up in the polar decomposition of a nonsingular matrix Z=UpH is a minimizer for both‖Log(Q⁎Z)‖and‖sym⁎(Log(Q⁎Z))‖ over the unitary matrices Q∈U(n) for any given invertible matrix Z∈Cn×n, for any unitarily invariant norm and any n. We prove that Up is the unique matrix with this property to minimize all these norms simultaneously. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.

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