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

Abstract

Let $H_n^{+}\left(R\right)$ be the cone of all positive semidefinite $n\times n$ real matrices. We describe the form of all surjective maps on $H_n^{+}\left(R\right)$, $n\ge 3$, that preserve the minus partial order in both directions.