How do you prove that

$\displaystyle \left\|A^{*} A \right\|_{2}= \left\| A \right\|^{2}_{2}$ ?

I can prove that $\displaystyle \left\|A^{*} A \right\|_{2} \leq \left\| A \right\|^{2}_{2}$

but I am not sure how to proceed for the other inequality.