Semidefinite Order on Symmetric Real Matrices

Let $k\ge1$ be a natural number and let $A$ and $B$ be symmetric real $k\times k$ matrices.

We write

if the difference $B-A$, formed entrywise — which is itself symmetric, since $(B-A)^{\top}=B^{\top}-A^{\top}=B-A$ with the transpose — is a positive semidefinite matrix. Here $0$ denotes the $k\times k$ matrix all of whose entries are $0$, so that $0\preceq A$ reads: $A$ is positive semidefinite.