For whatever reason, I have always defined matrices as being $n \times m$, and that is how I have been defining matrices throughout my dissertation. Recently however, I have noticed that nearly every other source primarily defines matrices as being $m \times n$. Is the later more formal notation? Should I go through my whole dissertation to change the notation? How important is it?
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1$\begingroup$ The point is not whether one says "n by m" or "m by n" but whether the first number counts rows or counts columns $\endgroup$– Yemon ChoiCommented Sep 16, 2021 at 20:29
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$\begingroup$ The convention I'm using is that $n \times m$ means $n$ rows and $m$ columns. $\endgroup$– PowerspawnCommented Sep 16, 2021 at 21:00
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$\begingroup$ Then I think that would be fine, although it might cause some readers a bit of confusion when they see an unexpected expression. $\endgroup$– Yemon ChoiCommented Sep 17, 2021 at 4:29
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2 Answers
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It is generally a good idea to keep things alphabetical, unless there is a good reason to do otherwise.
With matrices, it can be okay because an $n\times m$ matrix is really a linear transformation from $\mathbb{F}^m\to \mathbb{F}^n$ (where $\mathbb{F}$ is your field of definition), and so viewed this way, the $m$-dimensional space comes before the $n$-dimensional space.
So I think you are okay sticking with what you have.
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1$\begingroup$ Yes. But when it comes to variables for natural numbers, $n$ is the first letter of the alphabet. $\endgroup$ Commented Sep 16, 2021 at 17:58
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$\begingroup$ Nah. $i$ is a perfectly good natural number. :-p $\endgroup$ Commented Sep 16, 2021 at 18:08
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$\begingroup$ I also have defined vector spaces $U, V, W$ of dimensions $n,m,q$ respectively to define an $n \times m \times q$ tensor in $U \otimes V \otimes W$. Although tensors can also be interpreted as linear operators, it seems like this notation may be less acceptable. Would you agree? $\endgroup$ Commented Sep 16, 2021 at 19:10
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$\begingroup$ @Powerspawn It can really depend on a few factors, such as what is common in your field of study, etc... I'd recommend talking to your adviser. $\endgroup$ Commented Sep 16, 2021 at 19:16
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1$\begingroup$ As an example of a field-specific convention: in statistics the dimensions of a design matrix are traditionally referred to as $n \times p$, with $n$ the number of measurements and $p$ the number of parameters (variables): en.wikipedia.org/wiki/Design_matrix $\endgroup$ Commented Sep 17, 2021 at 0:21
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It is absolutely fine to do so.
Look at this. See how many authors/books have already used it.
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$\begingroup$ The point is not whether one says "n by m" or "m by n" but whether the first number counts rows or counts columns. $\endgroup$ Commented Sep 16, 2021 at 20:28
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$\begingroup$ In that case, in common parlance, it is rows first. $\endgroup$ Commented Sep 17, 2021 at 2:33