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.
But aren't all finite dimensional norms equivalent, so I can write my first formula? thank you.