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

## TL;DR

This paper characterizes all surjective maps on the cone of positive semidefinite matrices that preserve the L"owner and minus partial orders, with implications for statistical applications.

## Contribution

It provides a complete description of order-preserving maps on positive semidefinite matrices for both L"owner and minus partial orders.

## Key findings

- Surjective order-preserving maps for L"owner order characterized.
- Additive, order-preserving maps for minus order characterized.
- Results applicable to statistical contexts involving 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\"owner 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\"owner 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.