Monotone transformations on the cone of all positive semidefinite real matrices

Rok, strany: 2020, 733 - 744

Let $H_n⁺(\mathbb{R})$ be the cone of all positive semidefinite (symmetric) $n× n$ real matrices. Matrices from $H_n⁺(\mathbb{R})$ play an important role in many areas of engineering, applied mathematics, and statistics, e.g. every variance-covariance matrix is known to be positive semidefinite and every real positive semidefinite matrix is a variance-covariance matrix of some multivariate distribution. Three of the best known partial orders that were mostly studied on various sets of matrices are the L{ö}wner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on $H_n⁺(\mathbb{R})$ that preserve either the L{ö}wner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on $H_n⁺(\mathbb{R})$. We characterize surjective, additive maps on $H_n⁺(\mathbb{R})$, $n≥3$, that preserve the star partial order in both directions. We also investigate the form of surjective maps on the set of all symmetric real $n× n$ matrices that preserve the L{ö}wner partial order in both directions.

ISO 690:

Golubic, I., Marovt, J. 2020. Monotone transformations on the cone of all positive semidefinite real matrices. In Mathematica Slovaca, vol. 70, no.3, pp. 733-744. 0139-9918. DOI: https://doi.org/ 10.1515/ms-2017-0386

APA:

Golubic, I., Marovt, J. (2020). Monotone transformations on the cone of all positive semidefinite real matrices. Mathematica Slovaca, 70(3), 733-744. 0139-9918. DOI: https://doi.org/ 10.1515/ms-2017-0386

Vydavateľ: Mathematical Institute, Slovak Academy of Sciences, Bratislava