Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order

Abstract: Let $H_{n}<sup>{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. We describe the form of all surjective maps on $H_{n}^{{+}(\mathbb{R})}$, $n\geq 3$, that preserve the minus partial order in both directions.