# Math Help - Subordinate Matrix Norm equivalences.

1. ## Subordinate Matrix Norm equivalences.

Prove that for any matrix A and any vector norm ||·|| the following definitions of subordinate matrix norm ||A|| are equivalent...

||A|| = max(x!=0) ||Ax||/||x||

and

||A|| = max(||u||=1) ||Au||

I know I probably have to use the definitions of a matrix norm but I'm stumped. I don't really understand the topic all that well and none of the information I've found is very helpful.

Once again, any help greatly appreciated and sorry about the rubbish formatting.

2. $\sup_{V- \{ 0 \} } \frac{ \Vert Ax \Vert}{ \Vert x \Vert } = \sup_{V- \{ 0 \} } \Vert A( \frac{x}{ \Vert x \Vert}) \Vert = \sup_{S_V} \Vert Ax \Vert$ where $S_V$ is the unitary sphere of $V$. The first equality follows from the properties of the norm and the fact that $A$ is linear.