We have the Euclidean norm (or ) for a vector on the vector space ; .

Given a vector norm on the induced matrix norm for m x m matrices A is defined by

That is , is the smallest number such that

So given the Euclidian-norm for a vector the induced matrix norm

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 is symmetric.