The Structured Condition Number of a Differentiable Map between Matrix Manifolds, with Applications

Bahar Arslan*, Vanni Noferini, Francoise Tisseur

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

We study the structured condition number of differentiable maps between smooth matrix manifolds, extending previous results to maps that are only R-differentiable for complex manifolds. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure.

Original languageEnglish
Pages (from-to)774-799
Number of pages26
JournalSIAM Journal on Matrix Analysis and Applications
Volume40
Issue number2
DOIs
Publication statusPublished - 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • matrix function
  • Frechet derivative
  • condition number
  • bilinear form
  • sesquilinear form
  • structured matrices
  • structured condition number
  • automorphism group
  • Lie algebra
  • Jordan algebra
  • polar decomposition
  • DECOMPOSITION

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