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If I want to put emphasis on a matrix being entrywise-nonnegative, can I write in my paper

"...and thus there exists a nonnegative matrix $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$..."

or is that redundant and should be avoided?

is it instead better to say

"...and thus there exists a matrix $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$..."

Personally, I like the first choice, in order to put emphasis on the existence of a nonnegative matrix, but I wonder whether it's bad style for math writing.

I'm using $\mathcal{M}_N(\mathbb{R^+})$ to denote the set of $n\times n$ matrices with nonnegative entries.

Thanks,

User001
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    "is that redundant and should be avoided?" Not to comment on this particular case, but math papers are written for humans and redundancy is often good for communication. So I think your actual question is (or should be) only "Should this be avoided?", not "Is that redundant?" – JiK Sep 28 '16 at 08:38
  • What about "...and thus there exists a nonnegative matrix $\mathrm Q \in (\mathbb R_0^+)^{n \times n}$..."? – Rodrigo de Azevedo Sep 28 '16 at 08:47
  • @RodrigodeAzevedo: What good would that do? – tomasz Sep 28 '16 at 14:45
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    This has already been mentioned in both answers, but out of context, I would take the first example to mean a positive semi-definite matrix. – tomasz Sep 28 '16 at 14:48
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    @tomasz $\mathrm Q \geq \mathrm O_n$ may be read as "$\mathrm Q$ is positive semidefinite", but $\mathrm Q \in (\mathbb R_0^+)^{n \times n}$ is unambiguous. – Rodrigo de Azevedo Sep 28 '16 at 14:51
  • @tomasz Within a certain research area, "nonnegative matrix" is well understood and standard terminology. I have heard no one (apart from confused students) use the word "nonnegative" to mean "positive semidefinite". "Positive matrix" is more ambiguous, but this one is clear. Of course, in any case the first sections of the papers should contain an explanation of all the non-trivial notation and terminology used, including this one. – Federico Poloni Sep 29 '16 at 06:07

2 Answers2

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An issue with the first version is that it could indirectly suggests that not all matrices in that set are nonnegative. Then, somebody might wonder what "nonnegative" is supposed to mean. Could it mean the determinant is nonnegative or all eigenvalues or...?

This can create confusion and thus should be avoided. I agree though with the goal of putting emphasis on the fact, especially the first time it occurs. Consider this version or something along these lines instead:

...and thus there exists a matrix $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$, i.e., a matrix with nonnegative entries, ....

Put differently, redundancy can be good, but it can make sense to make clear that information is redundant.

quid
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    Awesome - thanks @quid :) – User001 Sep 28 '16 at 07:14
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    I like this solution, As a non-mathematician who often reads papers out of interest I can see myself scratching my head about the redundant explanation wondering if I've missed some aspect of how to describe matrices. This however makes it perfectly clear. – Robin Gertenbach Sep 28 '16 at 11:05
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    +1 "redundancy can be good, but it can make sense to make clear that information is redundant." – Dahn Sep 28 '16 at 13:41
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    I disagree. By the same token, your version could be taken to mean that not all elements of that set are matrices, and you should instead write "...and thus there exists a $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$, i.e., a matrix with nonnegative entries, ....". Personally, I would just drop the "$\in \mathcal{M}_N(\mathbb{R^+})$" part altogether. – tomasz Sep 28 '16 at 14:42
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    My preference is for "that is" instead of "i.e."; still, +1. – Greg Martin Sep 28 '16 at 18:20
  • @GregMartin actually mine too and I had written like this originally yet reverted in the grace period to get more distance to the other usage of "that is" like in "let $n$ be an integer that is squareful and divisible by three distinct primes", which I thought made sense given the context. – quid Sep 28 '16 at 18:48
  • @tomasz arguably it could be taken like this but I doubt anybody would take it like this while "nonnegative" actually is prone to misunderstanding as you yourself stressed on the post. Depending on what you want to do you cannot just drop it. Of course you could say let $Q$ be square matrix of dimension $N$ with nonnegative real entries, but you might not want to write this several times. Anyway, one certainly also could drop the "matrix" and it might be more consistent. . – quid Sep 28 '16 at 19:02
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    for sure, surrounding ", that is," by commas is crucial to avoid misunderstandings – Greg Martin Sep 28 '16 at 20:12
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First of all, I'm not sure "nonnegative matrix" means, unambiguously, "matrix with nonnegative entries." But my bigger issue with

...and thus there exists a nonnegative matrix $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$...

is that you may not need the notation at all. Sometimes programs that debug code will point out when you're declaring a variable but not using it for anything. It's not exactly an error, but it can complicate your writing.

If you're going to refer to the set of $N\times N$ matrices with nonnegative entries several times, name it at the beginning of a paragraph. Then use the shorthand notation later on. As in:

Let $\mathcal{M}_N(\mathbb{R^+})$ be the set of all $N\times N$ matrices with nonegative entries. ... yada yada yada ... and thus there exists $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$ such that ...

This way, someone who missed what exactly $\mathcal{M}_N(\mathbb{R^+})$ was can scan backwards to the beginning of the paragraph to find it. Someone who remembers can just move on without stumbling over an inline declaration.

If you're not going to refer to the set often, there's no need to name it with notation. Just say

... and thus there exists an $N\times N$ matrix $\mathcal{Q}$ with nonnegative entries such that ...

Erdos wrote “the best notation is no notation.” Use it only if you need it.

Matthew Leingang
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