Monotone transformations on the cone of all positive semidefinite real matrices

Abstract

<jats:title>Abstract</jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>(ℝ) be the cone of all positive semidefinite (symmetric) <jats:italic>n</jats:italic> × <jats:italic>n</jats:italic> real matrices. Matrices from <jats:inline-formula><jats:alternatives><jats:tex-math>$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>(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 Löwner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on <jats:inline-formula><jats:alternatives><jats:tex-math>$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>(ℝ) that preserve either the Löwner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on <jats:inline-formula><jats:alternatives><jats:tex-math>$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>(ℝ). We characterize surjective, additive maps on <jats:inline-formula><jats:alternatives><jats:tex-math>$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>(ℝ), <jats:italic>n</jats:italic> ≥ 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 <jats:italic>n</jats:italic> × <jats:italic>n</jats:italic> matrices that preserve the Löwner partial order in both directions.</jats:p>