Preservers of partial orders on the set of all variance-covariance matrices

Abstract

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L{\"{o}}wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on $H_{n}^{+}(\mathbb{R})$. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R})$, $n>1$, that preserve the L{\"{o}}wner partial order in both directions. We present an equivalent definition of the minus partial order on $H_{n}^{+}(\mathbb{R})$ and also characterize all surjective, additive maps on $H_{n}^{+}(\mathbb{R})$, $n\geq3$, that preserve the minus partial order in both directions.