
Euclidean Norm
We have the Euclidean norm (or $\displaystyle l_2norm$) for a vector on the vector space $\displaystyle R^m$ ; $\displaystyle \parallel{v}\parallel_2 = \sqrt{\sum_{j=1}^m\mid{v_j}\mid^2}$.
Given a vector norm $\displaystyle \cdot$ on $\displaystyle R^m$ the induced matrix norm for m x m matrices A is defined by
$\displaystyle A\ =\ max_{v\neq{0}}\frac{Av}{v}$
That is , $\displaystyle A$ is the smallest number $\displaystyle \alpha$ such that $\displaystyle Av\leq\alphav\ \forall\ v \in R^m$
So given the Euclidiannorm for a vector the induced matrix norm $\displaystyle A_2=\sqrt{max.\ eigenvalue A^TA}$
Can anyone proof/explain why?
In my book it is given but I donīt see it directly, i think it has something to do with the fact that $\displaystyle A^TA$ is symmetric.