How to prove that for any matrix $A$, $A^{T}A$ and $AA^T$ are non negative?

Question

I know that for any $A$, $AA^{T}$ and $A^{T}A$ are symmetric, so it's possible that these products are non-negative, where non negativity means that for a matrix $B$ to be non-negative, it has to yield $x^{T}*B*x\geq 0$ for any vector $x$. I could also see that if $\mid A^{T}A\mid=\prod\limits_{i=1}^{n}\lambda_{i}$, then given that $n=2k$ where $k$ is some natural number, its determinant would be positive and I know that $A^{T}A$ and $AA^T$ have the exact same eigenvalues. I'm not sure however, where to begin this proof or how to do this. Please help.