# A term for “number of columns” of a matrix

Is there a single English word to denote the "number of columns" of a matrix?

For example, "dimensionality" of a $2\times 3$ matrix is $2\times 3$. I need a term for $3$ in this example. I can always just say "number of columns", of course, but could I have a single word for it?

• Aha, interesting! In machine learning literature the convention is to use columns to represent features/variables, so perhaps something like this might apply in your case. – DataD'oh Nov 17 '17 at 11:16
• Can't you propose a context where you consider $m\times n$ matrices and then simply refer to $n$? – cangrejo Nov 17 '17 at 11:23
• I propose you do as Tukey would have and invent a word. How about "columnity" – einar Nov 17 '17 at 13:21
• @einar, unless we find a term that already exists, I would seriously consider your idea. I would not expect it to work immediately, but in the end we may recall that each useful term must have been invented at some point by people who could not do without it any longer. – Richard Hardy Nov 17 '17 at 13:36
• This is stretching it, but rank-nullity is a mathematical synonym for the number of columns: en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem – James Nov 17 '17 at 13:50

There is a concept of wide and narrow data, so maybe you could use the term „width“ for the number of columns after you define it in order to avoid the ambiguity.

• Thanks for your answer! This is close to what I need, but I wonder if there exists a more specific term for that. – Richard Hardy Nov 17 '17 at 11:56
• This is certainly a good idea. The frequent use of "tall and skinny" might also give you some clues. – cangrejo Nov 17 '17 at 11:59
• width is generally its name in computer programming, so +1 from me – BlueRaja - Danny Pflughoeft Nov 17 '17 at 14:50

1. You want a short, meaningful term.

2. You want it to be memorable and readable, rather than some clunky abstract mathematical or computerese construction like "let $\mathbb{A}\in\operatorname{Mat}(n,p)$" or even "$\mathbb{A}\in\mathbb{R}^{n\times p}.$"

3. You want to be able to specify the number of columns explicitly, so you can distinguish (say) 3-column matrices from 2-column matrices or $p$-column matrices from $p+1$-column matrices.

4. You want it function as a noun rather than an adjectival phrase; that is, it should read like "$\mathbb A$ is a $p$-column matrix" rather than "the column count of $\mathbb A$ is $p$."

5. Evidently, even a short phrase like "$p$-column matrix" is too much!

As others have remarked, you're in the domain of creating your own terminology. However, "columnity" (which has been proposed) has a grotesque and non-English aspect, albeit being little briefer.

If I were in this position, at the outset of the document I would introduce a term and define it, perhaps like this:

Because we will frequently need to refer to the number of columns in a matrix, let us say that a $p-$matrix is any matrix with exactly $p$ columns.

That seems to meet all the objectives. It's hard to imagine anything simpler, short of a mathematical notation (which violates objective $(2)$). It also is consistent with long-standing mathematical terminology, which includes well-known terms like "symmetric matrix," "real matrix," "transition matrix," "rank-$p$ matrix," etc.

• This is a good answer! I will think more and perhaps accept it. – Richard Hardy Nov 20 '17 at 7:20
• One problem with "p-matrix" is that it's ambiguous: "p" could refer to the number of rows just as easily. That's unless you've already specified that the matrix is nxp, in which case you would just refer to "p". And not everyone automatically makes matrices mxp. There's plenty of mxn matrices out there. For a general matrix term, it should be independent of application, so you can't assume that somehow the number of columns is more interesting. Both dimensions are important - you can't assume a reader will know you'll be talking about the number of columns instead of the number of rows. – gms Nov 24 '17 at 3:30
• @gms It's not in the least ambiguous if you follow the full prescription in this answer! – whuber Nov 24 '17 at 13:13
• @whuber, I realized that a sentence explains the convention. My point was that sentence would always be needed, because "p" doesn't strongly enough suggest columns rather than rows, rank, power, probabilities with rows or columns summing to 1, symmetric positive definite (my guess if I saw the the term without the explanation), "complex square matrix' (a Google result). The next author might just as easily have a sentence defining a p-matrix as one with p rows, or one with rank p. If the convention always requires the explanatory sentence, it fails the short and concise test. – gms Nov 24 '17 at 15:29
• @gms I do not interpret the question as asking for a way to create a new convention: I have only interpreted it as looking for a way to solve a particular problem of communication, as I outlined in the list of objectives that begin my answer. – whuber Nov 24 '17 at 15:48

Personally I would denote the matrix as $$X \in \mathbb{R}^{n \times p}$$ and use $p$ as a reference (assuming your matrix is composed of real values!).

Also note that the notation p >> n is quite widely used to describe the 'short and wide' datasets, e.g. datasets where the number of rows (observations) is significantly lower than number of columns (features). There is an area of Statistics known has 'High-dimensional statistics' that deals with these kinds of problems.

• Thanks for your answer! This is a nice idea for a workaround, and I am already using it. But this does not answer the question which specifically asks for a term (a single English word). – Richard Hardy Nov 17 '17 at 11:56
• Usually in Machine Learning and Statistics, each column corresponds to the dimensionality of the data or the number of covariates - I guess you could use these if you need a single word. I would recommend using what I suggested in my answer if you are writing a paper. – George Nov 17 '17 at 12:05
• George, as I said, I am already using it. Thanks again! – Richard Hardy Nov 17 '17 at 12:11
• Also also note that \gg typesets much nicer than >>. – David Richerby Nov 19 '17 at 1:44

I propose you do as Tukey would have done and invent a word. It is of course OK to define new terminology as long as we are explicit about it. As you say it might not gain immediate traction, but it would still work within the scope of your paper. My personal suggestion is

columnity [n.] of A: the extent to which A is columnar

An edit: Thinking a bit about my light-hearted suggestion here I felt I should also add that width is my preference. It's short and sweet and follows the excellent tip in Orwell's Politics and the English language to never use a long word where a short one will do.

• Surely you mean it to be a noun so he can say "of columnity 3"? (but +1 anyway) – mdewey Nov 17 '17 at 16:40
• I agree with @mdewey, I‘ve never heard columnity to be used as an adjective. But I‘m not native English speaker, maybe that’s why... P.S. you‘ve got my +1, of course. – aivanov Nov 17 '17 at 16:57
• @mdewey you are right of course, I was in a bit of a rush writing it down – einar Nov 17 '17 at 22:31
• That does sound a lot like "calumny" and "calamity". – Acccumulation Nov 18 '17 at 4:32
• @Acccumulation it's quite fortunate in a sense, a calamity being what you often get with too many columns – einar Nov 18 '17 at 5:59

I like "width", as suggested by aivanov. It is difficult to be more specific, without being longer and without using several words. If several words are ok, then we are back to "number of columns".

An alternative is to reformulate: Instead of saying:

$A$ is a matrix of width 3.

or

The number of columns of $A$ is 3.

why not just say

$A$ has 3 columns.

• Thanks for your answer! I do appreciate it, but it does not answer the question which specifically asks for a term (a single English word). If I needed ways of rephrasing, I would have asked for that specifically. – Richard Hardy Nov 17 '17 at 13:18
• @RichardHardy Well, I think you should give some examples of the sentences where you would like to use the term you searching for. It's a bit difficult to know what you want without a precise context. – amoeba Nov 18 '17 at 21:11
• @amoeba, see my comment under Glen_b's answer. – Richard Hardy Nov 20 '17 at 7:15
• @RichardHardy While the question you are technically asking is quite simple, it makes the appearance that there is an underlying question of style. Often, the most elegant solution is not the most direct solution. So don't complain if the proposed answers seem to go into different directions than you anticipate.. – jarauh Nov 22 '17 at 16:28
• Thank you for your comment. The question is phrased directly and the intention is that it makes the appearance of what it is. The question is hopefully precise enough to be understood in the same way by everyone. I have made no complaints so far and do not intend to make any (stating the fact is not a complaint). I appreciate the effort of everyone who has posted an answer, inlcuding yourself. It is worth noting that even the answers that do not answer the question may be useful for someone else with a related query at some point in the future; that is a nice thing to know. – Richard Hardy Nov 22 '17 at 22:01

It's not clear why it's important to have only a single-word term.

I'd say that for most purposes don't be afraid of three shortish words: "number of columns"; the very term you began with, which served to perfectly convey the idea you asked about.

It is a mere 5 syllables, it's not a strain to say or write.

A term consisting of just a few common words is likely to be less of an intellectual effort to grasp on the fly than one longish or unfamiliar one.

There may well be four or five terms that might serve, but if I was only writing it a few times, I would feel the strong inclination to go back and edit such a term out and replace it with "number of columns".

If I would need to write about a specific case many times, I'd start with something like "Let $X$ be an $n\times p$ matrix" and then refer to $p$ thereafter; or if I had to refer to the number of columns of many matrices, to have a notation like "denote the number of columns of a matrix $A$ as $c_A$", or something similar as suited the circumstances.

• My question was not about whether 3 words is better than 1, there was no intention to discuss that. The point was, does the term exist and if so, what is it. It is a pity a lot of discussion in this thread did not focus on the actual question. But thank you for your answer! – Richard Hardy Nov 20 '17 at 7:14
• Yes, I realize you ask for a single word and I don't give one; nevertheless, people looking for such a word who locate your post may be encouraged by my answer to use three instead. If I induce even a few such people to stick to three words for preference I think the answer will have been worth the effort. – Glen_b Nov 20 '17 at 7:20
• In that sense, you are of course right. Thanks again! – Richard Hardy Nov 20 '17 at 7:21

It seems like mostly a specific language (English) problem. In German language this would be much less problematic. The term 'spaltenanzahl' is not a strange word and regularly used. So you may consider introducing 'column-count' or 'column count' (while column-number would be ambiguous), or accept 'number of columns' as not that complex after all (and can often be written simpler with other sentence constructions e.g. 'matrix $M$ has $w$ columns').

Some background on the article and examples of sentences would help to look for different terms. One could use width and length or (as Sylvester, origin of the term matrix, did use) breadth and depth. Maybe based on what the matrix actually presents (e.g. system of equations, a polynomial, a vector space, etc.) other terms could be used.

Depending on your article background (or maybe no, independent from the public for your article, maybe only if you do something entirely new) I would advise to not use any newly invented term, and neither use some existing term (which must be an archaic term).

You have to ask yourself whether the poverty in the English language, not containing a simple synonym for the German term spaltenanzahl, is worth introducing something fancy that may only be confusing.

• Cool suggestions, breadth/depth has a cool ring to it as well. – Firebug Nov 19 '17 at 14:27

I always recommend avoiding to talk about matrices at all. Most applications that deal with matrices are in principle not interested in matrices at all, but rather in linear mappings between vector spaces. The basis-expanded representation of a mapping $\mathbb{R}^n \to \mathbb{R}^m$ is an $n\times m$ matrix, thus the column-count $n$ is the domain dimension of the mapping.

Likewise, the row count would be the codomain dimension.

These aren't single words, but they're reasonably short yet precise.

• This is an excellent suggestion from a mathematical point of view (+1). Since the most fundamental matrix considered in statistics is a data matrix, in which columns can have different data types, and such matrices cannot be conceived of as representing linear transformations of real spaces, the applicability of this suggestion is limited. – whuber Nov 19 '17 at 16:38

go for width \ height as mentioned its pretty clear what you mean and even a child knows what those words mean (providing the array is always represented from the same viewpoint)

Of course if you go into arrays of more than 3 dimensions (height, width, depth) it gets a little tricky and is probably better to use matrix notation

There does exist and English word columnarity which means the property of being columnar. So saying a matrix of columnarity 3 would seem quite natural.

• How is "a matrix of columnarity 3" better or simpler than "a matrix with 3 columns"? Sometimes it seems best to KISS. – user3697176 Nov 19 '17 at 13:15
• @user3697176 not when you want to refer to this very property. Compare given the number of columns of a matrix we can calculate [...] vs given the columnarity we can calculate – Firebug Nov 19 '17 at 14:26

I am guessing that you use your matrix for representing data. Usually, columns represent the different features and rows are different data points, as the data store in the figure below (ref).

This then extends to the the dimension of useful matrices, e. g. for mixing these features into a new dimension.

In that context, the number of columns is the number of dimensions of your feature space, that is, simply dimensionality.