The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure

Let $n$ be a natural number with $1\le n$, let $[n]$ be the initial segment determined by $n$, and let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure. Let $X$, $Y$ and $Z$ be symmetric real $n\times n$ matrices and let $\mu\in\mathbb{R}$. Write $X+Y$, $X-Y$ and $\mu X$ for the sum, the difference and the scalar multiple of matrices, and write $X\preceq Y$ for the positive semidefinite ordering.

Then the following hold.

1. (Symmetry is preserved) $X+Y$, $X-Y$ and $\mu X$ are symmetric.

2. (Reflexivity and transitivity) $X\preceq X$; and if $X\preceq Y$ and $Y\preceq Z$, then $X\preceq Z$.

3. (Translation) $X\preceq Y$ holds if and only if $X+Z\preceq Y+Z$.

4. (Nonnegative multiples) If $X\preceq Y$ and $0\le\mu$, then $\mu X\preceq\mu Y$.

5. (Agreement with the semidefinite order) $X\preceq Y$ holds if and only if $Y-X$ is positive semidefinite; that is, the ordering above and the one of Semidefinite Order on Symmetric Real Matrices agree.

6. (Antisymmetry) If $X\preceq Y$ and $Y\preceq X$, then $X=Y$.