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For a given matrix $\mathbf{A}\in\mathbb{C}^{m \times n}$, let $\|\mathbf{A}\| = \sum_{i=1}^m\sum_{j=1}^n|A_{ij}|$. Clearly, $\|\cdot\|$ is a matrix norm. Is there a special name and notation for $\|\cdot\|$? Well, name can be "entrywise $1$-norm" probably, but is there a conventional notation, e.g. $\|\mathbf{A}\|_{\mathrm{OMG}}$?

Lord Soth
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  • I guess you could call it the norm $1$ and denote it $|A|_1$. – Joel Cohen Apr 01 '13 at 18:23
  • @JoelCohen, Thanks for the reply, but $|\cdot|1$-norm appears to be reserved for $|\mathbf{A}|_1 \triangleq \max_j \sum{i=1}^m |A_{ij}|$ – Lord Soth Apr 01 '13 at 18:25
  • jupp the $|\cdot |_1$ norm is the column sum norm. nope i don't know a special name for that, if you take the squares and have the squareroot in the end it is the frobeniusnorm – Dominic Michaelis Apr 01 '13 at 18:29
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    The Wikipedia article on Matrix norms uses the notation suggested by @JoelCohen. The article notes: This is a different norm from the induced $p$-norm (see above) and the Schatten $p$-norm (see below), but the notation is the same. I don't know if this reuse of notation is widespread. – Jeppe Stig Nielsen Apr 01 '13 at 18:40

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While it is horrible to overload notation like this, the entrywise $p$-norm of a matrix is also unfortunately also denoted by $\|A\|_p$ in many books and such. Normally, though, I don't really see this notation come up very often. Most analyses are purely entry-wise, rather than trying to sum up the entry-wise errors. The only exception I've seen is $p = 2$, which is the Frobenius norm, which is nice because it is unitarily invariant.

One way to resolve the ambiguity is to denote the operator norm by $\|A\|_{p,q}$, where this quantity is the operator norm of $A$ as viewed as a map $(\mathbb{C}^n, \|\cdot\|_p) \rightarrow (\mathbb{C}^m, \| \cdot \|_q)$. In this case, the traditional operator $2$-norm of $A$ would be $\|A\|_{2,2}$. Of course, very few people do this, so until then we're stuck with ambiguous notation.

Christopher A. Wong
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  • Thank you, I guess you mean $|\mathbf{A}|_{p,q} \triangleq \sup{ |\mathbf{A}\mathbf{v}|_q:|\mathbf{v}|_p = 1}$, but unless I am missing something we cannot recover my OMG-norm from this definition, right? But the main point is taken, since there is no fixed convention, it appears that I am more or less free to choose my own notation; I may go with OMG :) – Lord Soth Apr 01 '13 at 18:51
  • Yes, the entry-wise norm is not an operator norm. – Christopher A. Wong Apr 01 '13 at 18:54