If A is invertible prove that B is positive definite.

Question

Let $A \in M_n(\mathbb{R})$ and let $B=A^TA \in M_n(\mathbb{R})$.

Prove that if A is invertible then B is positive definite.

So far I've done the following but I can't see how it helps. (https://i.stack.imgur.com/DXWBb.jpg)

Any help would be very much appreciated.

Answer 1

$B$ is positive definet if for every non zero $\mathbf v, \mathbf v^T B\mathbf v > 0$

$B = A^T A$

$\mathbf v^T B\mathbf v = \mathbf v^T A^T A \mathbf v = (A\mathbf v)^T(A\mathbf v)= \|A\mathbf v\|$

$\|A\mathbf v\| \ge 0$

If A is invertable, $\|A\mathbf v\| = 0$ only when $\mathbf v = \mathbf 0$