The Positive Semidefinite Ordering on Symmetric Matrices

Let $n\ge1$ be a natural number, and let $X$ and $Y$ be symmetric real $n\times n$ matrices. Let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure.

We write

if, with the dot product on Euclidean space $\mathbb{R}^n$ and the matrix-vector product,

for every $z\in\mathbb{R}^n$.