Show that $\text{trace}(A^TA) \ge 0$ and $\text{trace}(A^TA)$ if and only if $A = O$

Question

Problem

Show that $\text{trace}(A^TA) \ge 0$ and $\text{trace}(A^TA)=0$ if and only if $A = O$ when $A \in \mathbb{R}^{n \times n}, n \in \mathbb{N}$. Symbol $O$ denotes zero matrix.

Attempt to show

I think I know intuitively this is right but how to prove this? I could show some example but unfortunately, this won't exactly prove anything. The problem is how to approach this proof?

Answer 1

Hint: $\bf{A^{t}A}$ is a positive semidefinite matrix