Reading the wiki article I get confused about matrix norms. My question, is it true that $$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$ where $M$ is a matrix and $x$ is a vector and $C$ is some constant? What does $C$ depend on?
According to wiki, $\lVert Mx \rVert \leq \rho(M) \lVert x \rVert$ holds where $\rho$ denotes the spectral radius.
Yes, you can (the exact value of $C$ will depend on the particular choice of norm in the space and the matrix size). ---According to wiki, $\|Mx\|\le\rho(M)\|x\|$ holds where $\rho$ denotes the spectral radius--- This is nonsense as written. Where have you seen it? –  fedja Nov 15 '12 at 19:56