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Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results?

If they are identical, then I suppose the only difference between them is the method of calculation, eh?

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    What do you mean by 2-norm?2011-04-15
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    The p-norm where p=2, also known as the Euclidean norm.2011-04-15
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    If you mean the Euclidean norm when $M_n$ is treated like $\mathbb{C}^{n^2}$, then yes they are the same; this is the definition of the Frobenius norm, as seen on Wikipedia: http://en.wikipedia.org/wiki/Matrix_norm#Frobenius_norm2011-04-15
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    Ricket: Could you please give the precise definitions you are using (or precise references to these definitions)? The phrase "p-norm" for matrices is ambiguous, which is why I asked earlier. Many think of "2-norm" as meaning the operator norm when $M_n$ acts on $\mathbb{C}^n$ with Euclidean norm, hence Yuval's answer. I think I now know what you mean, but then your question is answered by the Wikipedia link, right?2011-04-15

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