# 2-norm of a matrix

• April 7th 2010, 01:44 PM
math8
2-norm of a matrix
How do you prove that

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

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

but I am not sure how to proceed for the other inequality.
• April 8th 2010, 02:25 AM
nimon
Which norm are you using for these matrices? Is it

$
\left\| A \right\|_{2} = \left( \sum\limits_{i,j} |A_{ij}|^{2} \right)^{\frac{1}{2}}
$

?
• April 8th 2010, 06:25 AM
math8
2-norm of a matrix
I think

$\left\| A \right\| _{2} = Sup_{x \neq 0} \frac {\left\| Ax \right\| _{2}}{\left\| x \right\| _{2}}$

and the one that you had might be the Frobenius norm.