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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.

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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
@fedja Thanks, good to know. so I can use the standard Euclidean norms. I thought I saw the spectral radius thing in wiki, but I guess not, but I found (…) where Fabian suggested something like it.. – maximumtag Nov 15 '12 at 21:10

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