Matrix Inequality and Higher Powers

Question

For arbitrary square matrices $M$, $N$ we write $M \geq N$ if $M − N$ is positive semi-definite.

If $M ≥ N$, can I say $M^2 \geq N^2$ ?

Answer 1

This is trivial, but lets close it out anyway.

Let $M=I_n,$ and $N=-2I_n,$ where $I_n$ is the identity. Then $M > N,$ but $M^2-N^2 = -3I_n,$ which is negative definite.