Euclidean Norm

**bram kierkels** · October 4th 2009, 07:03 AM

We have the Euclidean norm (or $l_2-norm$ ) for a vector on the vector space $R^m$ ; $\parallel{v}\parallel_2 = \sqrt{\sum_{j=1}^m\mid{v_j}\mid^2}$ .

Given a vector norm $||\cdot||$ on $R^m$ the induced matrix norm for m x m matrices A is defined by
$||A||\ =\ max_{v\neq{0}}\frac{||Av||}{||v||}$
That is , $||A||$ is the smallest number $\alpha$ such that $||Av||\leq\alpha||v||\ \forall\ v \in R^m$

So given the Euclidian-norm for a vector the induced matrix norm $||A||_2=\sqrt{max.\ eigenvalue A^TA}$
Can anyone proof/explain why?
In my book it is given but I don´t see it directly, i think it has something to do with the fact that $A^TA$ is symmetric.

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