Matrix Proof

**WWTL@WHL** · October 22nd 2008, 05:43 PM

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.

**ThePerfectHacker** · October 22nd 2008, 05:59 PM

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.

**WWTL@WHL** · October 22nd 2008, 06:11 PM

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?

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