By definition, an operator norm (a matrix norm induced by a vector norm) is submultiplicative.
I am wondering if the converse is true i.e. given a submultiplicative matrix norm, is there a vector norm which induces it?
Thank you!
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Frobenius. ${}$ – copper.hat Nov 23 '22 at 20:24
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2The Frobenius norm is a known example of a submultiplicative norm that is not an operator norm: for all operator norms, $|||I|||$ must be equal to $1$, but for the Frobenius norm $||.||_F$ we have $||I||_F = \sqrt{n}$ if $n$ is the dimension. – Bruno B Nov 23 '22 at 20:26
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You could also try the Frobenius norm. – copper.hat Nov 23 '22 at 20:35
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2@AnneBauval I was just joking, I mentioned the Frobenius norm in the first comment and it was repeated in the next, so I decided to go for a trifecta. – copper.hat Nov 23 '22 at 22:36
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But what about the Frobenius norm? – Ben Grossmann Nov 24 '22 at 19:02
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2@user See also my post here for some other examples, including most of the Ky-Fan and Schatten norms. – Ben Grossmann Nov 24 '22 at 19:03