This Wikipedia page defines the matrix $\ell_{p,q}$ matrix norm.
I was wondering what exactly the $\ell_{2,\infty}$ norm would be?
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1The formula on Wikipedia suggests that you get kind-of-a vector: $\tilde{a} = (\sum_{i=1}^m|a_{ij}|^2 )^{1/2}$ (operations are element-wise), after summing the inner part. After getting $\tilde{a}$, just take the infinity norm over it. That is $|A|_{2,\infty}=\max_j|\tilde{a}_j|$ for the above $\tilde{a}$. – o.spectrum Nov 22 '23 at 12:35
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1The logic is same for $\ell_{\infty, p}$ for some $p$, you just first get a vector of maximal elements from the rows of $A$, to which you apply the $p$-norm. – o.spectrum Nov 22 '23 at 12:38
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So the $\ell_{2,\infty}$ norm is the maximum column euclidean norm? – Dylan Dijk Nov 22 '23 at 12:40
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1Well, I guess, you can think of it like that, right – o.spectrum Nov 22 '23 at 12:41
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Ok thanks, I thought I heard in the past that it was rowwise instead. – Dylan Dijk Nov 22 '23 at 13:10
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Also, why is that when we take the limit this gives the infinity norm which is the max entry? – Dylan Dijk Nov 22 '23 at 13:24
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@o.spectrum So we can think of the inner part being a vector made of entries of the column euclidean norms and then applying the vector infinity norm, which as shown here is the max entry. – Dylan Dijk Feb 26 '24 at 15:52