# Math Help - Matrix Proof

1. ## Matrix Proof

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 ( $m \geq k$). Show that $A^T A$ is positive semidefinite.

My thoughts:

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

Any ideas?

Many thanks.

2. Originally Posted by WWTL@WHL
$x^T y \geq 0$, but didn't end up getting anywhere.

Any ideas?
.
Show that this is a sum of squares.

3. Originally Posted by ThePerfectHacker
Show that this is a sum of squares.
$x^T (A^T A) x = x^T ( \sum^k_{i=1} a^2_i ) x$

Does that hold?

Sorry, I'm quite clueless on this as you can probably tell. Any more guidance, please?