Transformations on positive definite matrices preserving generalized distance measures

L. Molnár, Patrícia Szokol

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

We substantially extend and unify former results on the structure of surjective isometries of spaces of positive definite matrices obtained in the paper [14]. The isometries there correspond to certain geodesic distances in Finsler-type structures and to a recently defined interesting metric which also follows a non-Euclidean geometry. The novelty in our present paper is that here we consider not only true metrics but also so-called generalized distance measures which are parameterized by unitarily invariant norms and continuous real functions satisfying certain conditions. Among the many possible applications, we shall see that using our new result it is easy to describe the surjective maps of the set of positive definite matrices that preserve the Stein loss or several other types of divergences. We also present results concerning similar preserver transformations defined on the subset of all complex positive definite matrices with unit determinant.

Original languageEnglish
Pages (from-to)141-159
Number of pages19
JournalLinear Algebra and Its Applications
Volume466
DOIs
Publication statusPublished - 2014

Fingerprint

Positive definite matrix
Distance Measure
Isometry
Non Euclidean geometry
Preserver
Unitarily Invariant Norm
Metric
Geodesic Distance
Divergence
Determinant
Unit
Subset
Geometry

Keywords

  • Distance measures
  • Jordan triple endomorphisms
  • Positive definite matrices
  • Preservers

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Numerical Analysis

Cite this

Transformations on positive definite matrices preserving generalized distance measures. / Molnár, L.; Szokol, Patrícia.

In: Linear Algebra and Its Applications, Vol. 466, 2014, p. 141-159.

Research output: Contribution to journalArticle

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