
2norm of a matrix
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.

Which norm are you using for these matrices? Is it
$\displaystyle
\left\ A \right\_{2} = \left( \sum\limits_{i,j} A_{ij}^{2} \right)^{\frac{1}{2}}
$
?

2norm of a matrix
I think
$\displaystyle \left\ A \right\ _{2} = Sup_{x \neq 0} \frac {\left\ Ax \right\ _{2}}{\left\ x \right\ _{2}} $
and the one that you had might be the Frobenius norm.