Hey, I can't figure out how to do this. I'd really appreciate any help.

Question:

A is a non-zero matrix of order m by k ($\displaystyle m \geq k$). Show that $\displaystyle A^T A$ is positive semidefinite.

My thoughts:

It's quite easy to show that $\displaystyle A^T A$ is symmetric, and I tried to use that fact and let $\displaystyle y = Ax$, and I tried to show $\displaystyle x^T y \geq 0 $, but didn't end up getting anywhere.

Any ideas?

Many thanks.