### 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 language | English |
---|---|

Pages (from-to) | 141-159 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 466 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 466, pp. 141-159. https://doi.org/10.1016/j.laa.2014.09.045

}

TY - JOUR

T1 - Transformations on positive definite matrices preserving generalized distance measures

AU - Molnár, L.

AU - Szokol, Patrícia

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Distance measures

KW - Jordan triple endomorphisms

KW - Positive definite matrices

KW - Preservers

UR - http://www.scopus.com/inward/record.url?scp=84908426266&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908426266&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2014.09.045

DO - 10.1016/j.laa.2014.09.045

M3 - Article

AN - SCOPUS:84908426266

VL - 466

SP - 141

EP - 159

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -