Talk:Matrix (mathematics)

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	Matrix (mathematics) has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
Article milestones Date Process Result April 27, 2009 Good article nominee Listed June 21, 2009 Peer review Reviewed
Current status: Good article

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Inconsistency -- what is a matrix?

The definition is unclear: in the very first sentence it is just a way of representation -- mathematical quantities in a rectangular array. In this sense, a calendar sheet that shows the dates of a month arranged by weeks would also be a matrix. Later comes the statement that you can add or even multiply matrices, which goes beyond that. Then, it again says that "major application of matrices is to represent linear transformations" (should probably read linear map), so if this is just the major application, calendar sheets would indeed fall into the category matrix. But then below under the heading "Definition" addition and multiplication are again required, and essentially all the rest of the article is about computations on matrices. Historically, Sylvester's introduction of the term also is only in the context of computability. I would argue to restrict the meaning of matrix here to those rectangular arrays of quantities that at least allow meaningful matrix multiplication, and I think that I am in line with most textbooks on that. Specifically, the introduction should reflect that explicitly. What are your thoughts on that? Seattle Jörg (talk) 07:39, 27 July 2021 (UTC)[reply]

Things are not as simple as it could appear: with your suggestion, an incidence matrix would not be a matrix. So, I suggest to expand the first sentence as follows, and to upgrade the article accordingly:

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, that is used to represent a mathematical object or a property of such an object. Generally, the operations on the represented objects are reflected by corresponding matrix operations. Without further specifications, matrices represent linear maps; their scalar multiplication, addition and multiplication correspond to scalar multiplication, addition and composition of linear maps.

By the way, the current lead is much too long, and contains to much technical details that belong to the body. IMO, the lead must be reduced to: the preceding quotation (or a variant of it); a short paragraph linking to other kinds of matrices and stating that the remainder of the article is about the matrices of linear algebra; a paragraph on square matrices; a paragraph on computational linear algebra and applications outside mathematics (this may be in the same paragraph, as most applications outside mathematics use computers). D.Lazard (talk) 09:16, 27 July 2021 (UTC)[reply]

I have rewitten the lead for fixing this issue. I have also removed many technical details that do not belong to the lead, for getting a lead of a reasonable length. The article body still requires to be updated, in particular for inserting in it details that I have removed from the lead, which were not duplicated in the body. D.Lazard (talk) 10:15, 28 July 2021 (UTC)[reply]

That's definitely an improvement, thank you. Seattle Jörg (talk) 11:10, 28 July 2021 (UTC)[reply]

No, it is not right. There are multiple problems. First, "without further specification" as no meaning. Can we say "here is a matrix of numbers but you are forbidden to multiply it by a vector"? Actually, mathematics is full of examples where matrices arise from a non-linear-algebra context but are then analyzed using linear algebra. Two examples are provided but claimed to be examples of the opposite: in combinatorics, adjacency and incidence matrices are defined as properties of discrete structures but there is a large industry of doing linear algebra with those matrices to analyze those structures. See spectral graph theory for one. An example of a combinatorial matrix which is rarely (but not never) regarded as a linear map is a Latin square. What the article can honestly report is that the most common use of a matrix in mathematics is to represent a linear map, and then immediately give an example (say ${\displaystyle 2\times 2}$) to show what that means. Currently this use of a matrix is not even defined until much later in the article. Another thing: when a textbook like Lang defines "matrix" they are telling you what meaning the term has in the book. It doesn't mean that Lang would deny that, say, a Latin square is a matrix but only that it is out of scope in the context. It is different for an encyclopedia. McKay (talk) 04:37, 29 July 2021 (UTC)[reply]

The formulation "without further specification" can certainly be improved. The intended meaning if that, when one encounters the word "matrix" without any specification of the kind of matrix that is considered, this is in relation with linear algebra. This does not deny that other rectanguar arrays are called matrices (examples are given in the same paragraph). This does not deny either that these other matrices may have hidden relations with linear algebra (examples given in a footnote). IMO, the fact that, by default, a matrix represents a linear map, is important enough for appearing soon and clearly in the lead. By the way, your example of Latin squares is not really convenient here, as Latin squares are rarely called matrices, at least in Wikipedia article. D.Lazard (talk) 09:49, 29 July 2021 (UTC)[reply]

Latin squares are one of my specialties and I'd be very surprised if any of my colleagues denies that they are a type of matrix. But I'm not proposing they be mentioned in the lead. The major problem is that the lead says "Without further specifications, matrices represent linear maps" but not a clue is provided as to what that means. The poor reader has to find their way down to the "linear transformation" section and try to decode the explanation given there. McKay (talk) 08:00, 5 August 2021 (UTC)[reply]

I agree, this should be changed. And the lead contains details that should be left to the main text. --Andres (talk) 22:11, 8 October 2023 (UTC)[reply]

I've tweaked the wording to avoid the (unclear, at least to me) phrase "without further specification". --JBL (talk) 23:29, 8 October 2023 (UTC)[reply]

Format - matrix example

Thanks everyone for your input to create this. I believe the matrix example near the top should show the subscripts for the first two columns to be: a11, a21, a31, am1 and a12, a22, a32, am3 (the row subscripts are not indexing on the example). The last column follows the proper format. CarmenRx (talk) 13:08, 10 September 2023 (UTC)[reply]

This is possibly a problem of your viewer. On my laptop and my i-phone, the column indices are displayed correctly (in red). D.Lazard (talk) 15:34, 10 September 2023 (UTC)[reply]

Good call D.Lazard. It is the correct on my laptop (but not on my iPhone). CarmenRx (talk) 18:03, 10 September 2023 (UTC)[reply]

Numbers

In the first sentence, we read: ... table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object .... But numbers are mathematical objects. Some other wording is needed. English isn't my first language, so I cannot propose any better wording. --Andres (talk) 19:54, 8 October 2023 (UTC)[reply]

Symbols and expressions are also mathematical objects. I would suggest table of numbers, or other mathematical objects, arranged in rows and columns, which is used to represent a mathematical object that is associated to all entries of the table, but I am not sure that it is easier to understand. D.Lazard (talk) 20:05, 8 October 2023 (UTC)[reply]

The point is that objects and their signs (that are mathematical objects only qua types, not qua tokens) have to be distinguished. The objects themselves don't form any rectangle, their sign tokens do. Anyway, "rectangular array" isn't clear enough. Matrices aren't literally data types. --Andres (talk) 22:05, 8 October 2023 (UTC)[reply]

Essentially every reliable source on matrices uses the phrase "rectangular array" or equivalent, there is no possibility of it not appearing early and prominently in any discussion of what a matrix is. --JBL (talk) 23:36, 8 October 2023 (UTC)[reply]

Then this needs more explication. "Array" is linked to Array (data type). I don't think we should define matrices via data types. --Andres (talk) 23:43, 8 October 2023 (UTC)[reply]

I agree: "array" is used here in its simple common language meaning. I have removed the wikilink. --JBL (talk) 00:22, 9 October 2023 (UTC)[reply]

which is used to represent a mathematical object or a property of such an object

Matrices are mathematical objects on their own right. --Andres (talk) 10:05, 9 October 2023 (UTC)[reply]

Infinite matrices and empty matrices

It seems to me that neither infinite matrices nor empty matrices can be subsumed under the official definition, so the presentation is inconsistent. By the way, there is another generalization: hypermatrix. --Andres (talk) 22:16, 8 October 2023 (UTC)[reply]

It is true that infinite matrices are not matrices under the usual definitions, and also true that for many purposes it is sensible to class them with and study them alongside matrices. Roughly the same applies to matrices with 0 rows or 0 columns or both (though in some contexts they may in fact be admitted as matrices). Natural language is not a rigorous system, even in the context of natural language in mathematics.

Hypermatrices are mentioned briefly in the article, but under the alternative name "tensors". --JBL (talk) 23:34, 8 October 2023 (UTC)[reply]

This seems trivial to experienced mathematicians but I'm pretty sure that for beginners, lack of rigour can become an obstacle of understanding. Andres (talk) 23:46, 8 October 2023 (UTC)[reply]

It is not possible to develop, in an encyclopedic context, a rigorous treatment of the word "matrix" that is consistent across all uses, because those uses are not consistent with each other; this is not a flaw in the way the Wikipedia article is written, it is just the way life is. "Infinite matrix" is a very clear example of this, but so are thousands of other things; a common example is that "surfaces with boundary" are not surfaces under the usual definition. (Allowing myself to follow you in an off-topic direction: in my experience as an educator, beginners need to develop intuition before they can develop rigor, not the other way around.) --JBL (talk) 00:29, 9 October 2023 (UTC)[reply]

I agree that intuition should be developed before rigour. But once rigor begins, it should be really rigorous. Only when rigour is acquired, loose expressions become acceptable. So, then the article needs an intuitive introduction before the definition. I admit this would be the hardest part of writing the article. Andres (talk) 00:43, 9 October 2023 (UTC)[reply]

And oriented graphs are not graphs. I think this is no problem if this is clearly stated. --Andres (talk) 00:48, 9 October 2023 (UTC)[reply]

Of course they belong to the topic but I think it should be clearly and precisely stated how they relate to usual matrices. Andres (talk) 23:50, 8 October 2023 (UTC)[reply]

You are of course welcome to edit the article, or to make concrete proposals for changes. --JBL (talk) 00:30, 9 October 2023 (UTC)[reply]

Yes, maybe I will. --Andres (talk) 00:48, 9 October 2023 (UTC)[reply]

Subtraction

Under "basic operations" subtraction isn't mentioned (it is twice under generalizations). Matrix subtraction is a redirect to Matrix addition. But there subtraction isn't mentioned. Subtraction doesn't reduce to addition when we have no operation of opposite (opposite element, that is, inverse element as to addition). So we should have either subtraction or opposite among basic operations. --Andres (talk) 00:56, 9 October 2023 (UTC)[reply]

In the case of real and complex numbers the opposite is multiplication by –1. I don't know ho far this can be generalized. In any case, this should be mentioned, I think. --Andres (talk) 01:01, 9 October 2023 (UTC)[reply]

I have added subtraction, with other cosmetic changes. D.Lazard (talk) 11:40, 9 October 2023 (UTC)[reply]

"Matrix Theory and Linear Algebra" listed at Redirects for discussion

The redirect Matrix Theory and Linear Algebra has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 November 15 § Matrix Theory and Linear Algebra until a consensus is reached. Steel1943 (talk) 20:58, 15 November 2023 (UTC)[reply]

"Matrix(mathematics)" listed at Redirects for discussion

The redirect Matrix(mathematics) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 November 15 § Matrix(mathematics) until a consensus is reached. Steel1943 (talk) 21:00, 15 November 2023 (UTC)[reply]

Article review

It has been a while since this article has been reviewed, so I took a look and saw lots of uncited statements. While some of the statements could be cited to the mathematical data included in the article, other prose (such as in the "History" and "Electronics" sections) needs to be cited. Should this article go to WP:GAR? Z1720 (talk) 15:25, 24 April 2025 (UTC)[reply]

GA Reassessment

Matrix (mathematics)

Article (edit | visual edit | history) · Article talk (edit | history) · Watch • Watch article reassessment page • Most recent review

Result pending

While some of the statements could be cited to the mathematical data included in the article, other prose (such as in the "History" and "Electronics" sections) needs to be cited. Z1720 (talk) 05:56, 3 May 2025 (UTC)[reply]

There were not before the GAR was opened and still are not any cleanup tags on this article such as [citation needed]. The only actual error category on the article is the newly-instantiated ISBN-date mismatch. Therefore, to me, starting the cleanup process by immediately opening a GAR seems like an excessively strong first measure. Maybe it would have been less confrontational to have tried placing [citation needed] tags first, and then waited enough time to see whether they were addressed, disputed, or ignored before opening a GAR? Even now that the GAR has been opened, you could still place those tags. Doing so would make it more clear to editors what you think is inadequately cited rather than this vague wave which leaves much to the imagination and makes it impossible to determine whether any steps one might take would satisfy you.

To put it another way: the preferred outcome of a GAR is to restore an article to deserving GA status, not to delist. If cleanup tags can head off a GAR before it starts, that would be even better. And telling us that the article is inadequate without providing specific-enough guidance for why you think so is a step towards the non-preferred outcome. —David Eppstein (talk) 07:38, 3 May 2025 (UTC)[reply]

@Z1720 can you provide some citation needed tags and other tags for describing the problem you have listed? As for a quick note, I am pinging @Jakob.scholbach as the nominator in 2009. Dedhert.Jr (talk) 07:52, 3 May 2025 (UTC)[reply]

• @David Eppstein: Some editors have described the addition of citation needed templates as disruptive, which is why I only add them when asked. I have now added some where I think they are needed and I see that other editors have also added cn templates to the article. If any editors are concerned about my conduct in GARs, please open a new thread at WT:GAN where the conversation may be more appropriate. Z1720 (talk) 15:47, 3 May 2025 (UTC)[reply]

  I did open a thread about WT:GAN, not about you specifically but about whether it is reasonable to expect that an article not tagged for any problems to suddenly come under GAR. And in that thread, you deflected again, saying that you would rather be pinged than discuss things there. —David Eppstein (talk) 17:59, 3 May 2025 (UTC)[reply]

Unfortunately, Jakob.scholbach appears not to have been active for almost a year. Fortunately, matrices are a basic enough topic that any other mathematician should be able to contribute, without requiring any special expertise. —David Eppstein (talk) 18:01, 3 May 2025 (UTC)[reply]

Anyway, lest this GAR appear inactive: improvement to the tagged missing citations has been ongoing on the article itself. —David Eppstein (talk) 17:49, 6 May 2025 (UTC)[reply]

Indeed, I was inactive on Wikipedia for some time, but by accident I stumbled across this GAN. I am not convinced this GAN is actually warranted, but I will try to allocate some time to resolve the citation needed tags. Any help is of course appreciated! Jakob.scholbach (talk) 14:45, 8 May 2025 (UTC)[reply]

@Jakob.scholbach Since the nominator, again, did not provide further comments, I should have intervened as well. @Z1720, I will take over the nomination, hope you do not mind. Dedhert.Jr (talk) 00:52, 9 May 2025 (UTC)[reply]

@Dedhert.Jr: Happy for anyone to help. Feel free to ping me if you have any questions or if this is ready for a re-review. Z1720 (talk) 01:00, 9 May 2025 (UTC)[reply]

Some comments:

• I never heard of ${\displaystyle M}$, or ${\displaystyle \operatorname {Mat} }$, is used in place of ${\displaystyle {\mathcal {M}}}$ as a symbol for square matrix, although it is used in some StackExchange's posts. If this is often, then more sources are preferable use them at all. Dedhert.Jr (talk) 03:25, 9 May 2025 (UTC)[reply]
• The article is getting technical as scrolling down, starting from Linear transformations. Another one is in the infinite matrices, where ${\displaystyle \bigoplus }$ is not very well-known to strangers.

I agree this section was not well done. I have removed most of it. Jakob.scholbach (talk) 19:48, 10 May 2025 (UTC)[reply]

• "There are many algorithms for testing whether a square matrix is invertible." Should you add some more algorithms?
• I will add some citation needed tags in some places, but I will also help to supply the requested citations.

That's all, and I'll check again after this. My time is short now. Dedhert.Jr (talk) 03:25, 9 May 2025 (UTC)[reply]

The first paragraph of the "Matrix groups" section has a confusing (and maybe redundant?) footnote that refers to "the general linear group" before general linear groups are defined. Footnote 95 just says "See any reference in representation theory or group representation." As I understand the culture here, that should be replaced with a specific book.

I think it is fine this way, especially given that is just a footnote. There is a tradeoff between keeping the focus and being correct (or even pedantic) here. Jakob.scholbach (talk) 19:48, 10 May 2025 (UTC)[reply]

The section heading "Linear combinations of quantum states" seems rather obfuscated. Why not just call it "Quantum physics" or "Quantum mechanics"? The text is also somewhat confused. Density matrices aren't an example of "matrix mechanics" as Heisenberg developed it in 1925; they were introduced some years later. 64.112.179.236 (talk) 08:58, 10 May 2025 (UTC)[reply]

OK, I changed the quantum physics paragraph. I think bringing in "eigenstates" was also confusing. That was just one more unfamiliar and undefined term. 64.112.179.236 (talk) 09:20, 10 May 2025 (UTC)[reply]

Are there rules for what goes in the "See also"? It looks kind of like a junk drawer. I mean, why "Bohemian matrices", of all things? 64.112.179.236 (talk) 19:28, 10 May 2025 (UTC)[reply]

I agree, these items should mostly (or all?) be removed.Jakob.scholbach (talk) 19:48, 10 May 2025 (UTC)[reply]

It's been trimmed now; I think the present status looks OK. 64.112.179.236 (talk) 21:37, 11 May 2025 (UTC)[reply]

The footnote to Mehra & Rechenberg (1987) needs a page number. Those are pretty big books. 64.112.179.236 (talk) 19:36, 10 May 2025 (UTC)[reply]

The ISBN given was for the wrong volume (Part 1 of Volume 5 is about Schrödinger's wave mechanics, not the Heisenberg–Born–Jordan matrix mechanics). I replaced it with a reference to the relevant pages in B. L. van der Waerden's editorial introduction to Sources of Quantum Mechanics, which is probably easier to get a hold of anyway. 64.112.179.236 (talk) 21:53, 11 May 2025 (UTC)[reply]

I have added a few references in response to the citation needed tags. My overall impression is that the large majority of these requests are quite overblown: especially when it comes to the more survey-like sections, a look in the corresponding sub-article will practically always bring up references etc. Jakob.scholbach (talk) 19:48, 10 May 2025 (UTC)[reply]

• When I saw some of the cn tags removed on my watchlist, I checked and agreed that they could be removed. If I disagree, I'll post below. Regarding the latter part of your comment about "corresponding sub-article will practically always bring up references": Information in the article needs to be referenced in the same article, as Wikipedia does not expect readers to click on a wikilink and find the information in a sub-article to verify the information (WP:V). Of course, Wikipedia articles can use the same sources: if the source is in the sub-article, the referencing can be copied and pasted into this article. Z1720 (talk) 19:56, 10 May 2025 (UTC)[reply]

I have resolved on more cn tag (which again, IMO, was pointless), and have asked the Chemistry and Physics "departments" for help with two others. Jakob.scholbach (talk) 07:26, 20 May 2025 (UTC)[reply]

I replaced the statements in the 'Electronics' section with something general, but at least it serves to point to the application. The previous content was logically incomplete (when would one multiply matrices?) and likely incorrect not matter how one fills in the blanks. Hopefully we do not need a more specific example here for a GA. —Quondum 18:07, 20 May 2025 (UTC)[reply]

I do see that this GAR has stalled over the last 10 days. I would like to ask where the consensus is on whether the article still meets the good article criteria now that edits have been made to address the concerns made in this GAR? Gramix13 (talk) 01:01, 30 May 2025 (UTC)[reply]

@Gramix13: Editors are still making changes to the article, and this has been ongoing for several days: when the changes are complete I am happy to re-review if pinged. I think this can stay open for a bit longer. Z1720 (talk) 12:26, 30 May 2025 (UTC)[reply]

Object or Expression?

The lead seems to give the impression that a matrix is a kind of expression which seems to contradict how it is used body which uses it as a specific mathematical object. I don't think it is controversial to say a matrix is the object, not the table (For example, ${\displaystyle A\in \mathbb {R} ^{3\times 4}}$ defines a matrix, A, even though "A" is not a table of numbers). I'd like to change the lead paragraph to something more like:

"In mathematics, a matrix is a mathematical object represented by a rectangular array or table of numbers, symbols, or other expressions, with elements or entries arranged in rows and columns, satisfying certain rules of addition and multiplication."

Does anyone have an issue with this? – Farkle Griffen (talk) 22:39, 19 June 2025 (UTC)[reply]

In principle this is ok with me, but there is a small issue here (in some ways too technical for the lead): when we talk about matrices as mathematical objects, we need to distinguish them from other sort-of-the-same mathematical objects, linear transformations and two-dimensional tensors. A matrix really is the table of numbers, while those other things can be converted into tables of numbers once you have a basis but don't need the basis to do what they do.

For this reason I would rather say "consisting of a rectangular array..." rather than "represented by a rectangular array...". There is no other more-abstract thing that we are representing by the array. The thing is the array. —David Eppstein (talk) 23:23, 19 June 2025 (UTC)[reply]

The distinction I want to make is between meta-mathematics and mathematics, i.e, the thing you draw and the thing it represents. The difference here is the array contains symbols, whereas the matrix contains objects. Further, the purely mathematical matrix doesn't really care "where" the objects are, so long as it can answer questions about "rows", "columns", and "entries" the way we expect. As opposed to a literal array where the entries must be in literal columns and rows.

I suppose if you define "rectangular array" abstractly, as a kind of mathematical object, then the issue goes away, but I think that would probably be too confusing. – Farkle Griffen (talk) 23:54, 19 June 2025 (UTC)[reply]

I suppose it could be defined as a function ${\displaystyle [1,n]\times [1,m]\to \mathbb {R} }$ (or whatever other codomain one wants) but even if we could source that it would probably be more confusing than helpful; the people who would understand it that way already know what a matrix is. I think more often when I've seen it formally defined as something like this it is as a tensor for a tensor space with a specified basis, even less helpful for the same reason. —David Eppstein (talk) 00:12, 20 June 2025 (UTC)[reply]

Or a list of lists, like [[1,2], [3,4]] which is common in computer science. I believe in set theoretic-constructions, tuples are sometimes defined as functions, ${\displaystyle \{1,2,...,n\}\to S}$, and matrices as a tuple of tuples (thus ${\displaystyle \{1,2,...,m\}\to (\{1,2,...,n\}\to \mathbb {R} )}$), so I don't think it would be too hard to find a source that defines matrices similar to what you mentioned.

In any case, "Matrix" is essentially all of these. That is, it is just any "mathematical object that behaves like an array", the exact construction doesn't usually matter, which I think my lead proposal reasonably summarizes without being too confusing. – Farkle Griffen (talk) 00:30, 20 June 2025 (UTC)[reply]

In computing very often a matrix is just a big block of (linearly indexed) memory with some metadata attached about the size and some special code written to convert two-dimensional lookups to one-dimensional ones. Sometimes there's extra padding (to the nearest multiple of some power of two) between rows because that speeds up SIMD code or improves cache performance. Occasionally the data is stored in z order. Sometimes in higher-level programming languages a matrix is represented as a list of lists; performance is then usually much worse, but in some contexts it's not a relevant bottleneck or the programmer doesn't care. I don't think "anything that behaves like an array" is really essentially different from "array" when "array" hasn't already been given a specific technical definition, and trying to draw a distinction seems like a distraction. –jacobolus (t) 00:58, 20 June 2025 (UTC)[reply]

Sure, but by this article's definition, none of these definitions above would be matricies since none of them are "a rectangular array or table of numbers, symbols, or expressions" – Farkle Griffen (talk) 01:03, 20 June 2025 (UTC)[reply]

Quite the opposite, all of these are (computer representations of) tables (i.e. 2D arrays) of numbers. What makes something a "matrix" is what kind of API you build around it and how you think of it conceptually, rather than how the data is laid out in memory. –jacobolus (t) 01:11, 20 June 2025 (UTC)[reply]

Completely agree. Similarly, the set-theoretic constructions are (mathematical representations of) tables of numbers. None of which are necessarily "rectangular" nor do they contain mathematical expressions or symbols. None of them fit the definition given without assuming some abstract meaning of the very unassuming term "rectangular array". Hence, "behaves like a rectangular array" rather than literally, a rectangular array. – Farkle Griffen (talk) 01:24, 20 June 2025 (UTC)[reply]

As David said above, "people who would understand it that way already know what a matrix is" (and will be able to imagine whatever pedantic distinctions they need for some fancier context), while people who do not know what a matrix is are going to be more confused/distracted than helped. Try to imagine yourself as a high-school student or curious layperson, and consider whether this distinction would be meaningful or informative. In my opinion to really flesh it out meaningfully is going to take multiple paragraphs of preliminary definitions and context, none of which is frankly all that relevant to the lead here. If you really think this needs belaboring, consider adding a footnote somewhere in § Definition. –jacobolus (t) 01:32, 20 June 2025 (UTC)[reply]

It's a pretty small change. As a high-school student, I don't think I would notice the difference between "Is a rectangular array" and "is represented by a rectangular array", and would probably glaze passed it, getting the same message.

Why is that any reason to not correct the article? WP:Technical makes it pretty clear that you shouldn't lie to children.

(Edit: This response was made before your edit) – Farkle Griffen (talk) 01:39, 20 June 2025 (UTC)[reply]

I don't think this is a "lie to children". All it does is leave out a set of narrowly pedantic and in context unnecessary definitions that might conceivably be inconsistent; but if we pick different definitions there is no inconsistency and no problem (and indeed under alternate definitions adding "represented by" would become inconsistent). A Google scholar search for "matrix is a table" turns up ~7000 results, and "matrix is an array" turns up 1000, "matrix is a rectangular" turns up 1600, "matrix is a two-dimensional" turns up 2300, etc. Or in short, it is standard, correct, and unambiguous to say that a matrix is a table or array of numbers or other objects. –jacobolus (t) 02:50, 20 June 2025 (UTC)[reply]

And "matrix is represented by" returns 13,000 results, and "matrix is denoted by" returns 24,000. Sheer numbers on a snippet of text don't mean much taken out of context. In context, matrices are always used as objects, but in this article, they are presented like a kind of notation. Even you said "array of numbers or other objects." but the article clearly says "array of numbers, symbols or other expressions", which disagrees. If you'd like to leave out the term "mathematical object," that's fine, but the current article confuses matrices with their notation. – Farkle Griffen (talk) 03:10, 20 June 2025 (UTC)[reply]

The language confuses matrices with their notation. In English, when we talk about a table of numbers, we might mean a mathematical object, or we might mean a graphically presented array of digit sequences. It is hardly the only kind of metonymy where we use the same words to talk about a thing and its appearance. This happens even for numbers, where far too often people confuse numbers as mathematical objects with their decimal representations (a big part of why people have problems with 0.999...). —David Eppstein (talk) 05:38, 20 June 2025 (UTC)[reply]

I agree, and that's fine, but it should be clear which of the two is the subject of this article. We don't merge number and numeral system just because people conflate the terms. – Farkle Griffen (talk) 06:18, 20 June 2025 (UTC)[reply]

The subject of the article is matrices, rectangular arrays of entries (which can be numbers or other objects). A matrix is a type of mathematical object. The subject is currently clear and there's no significant confusion about it. I really don't understand what difficulty you are having here. I think you have some idiosyncratic / personal definitions which don't accord with the mainstream.

If you look up "matrix" in pick-your-favorite mathematical encyclopedia you are going to find something like "a matrix is a rectangular array of ...". –jacobolus (t) 06:24, 20 June 2025 (UTC)[reply]

What, exactly, do you think I'm objecting to? What "personal definitions" do you think I have? – Farkle Griffen (talk) 06:35, 20 June 2025 (UTC)[reply]

You seem to think that the words "table" or "array" can't refer to an abstract object. It's not clear why. –jacobolus (t) 07:38, 20 June 2025 (UTC)[reply]

This reminds me of a discussion I read somewhere about a workbook for schoolchildren at the age where they're learning how to follow directions and use crayons. The old version of the book said, "Color the dog brown." The new version changed this to say, "Color the picture of the dog brown." It's not a real dog, you see, just an outline on paper, and we shouldn't confuse the two. But the picture of the dog was a rectangle with a background, like some squiggles representing trees. Does "color the picture of the dog brown" mean filling in the whole rectangle with the brown crayon? Pedantically "clarifying" a point about which no actual children were confused ends up losing on its own terms. Stepwise Continuous Dysfunction (talk) 15:59, 20 June 2025 (UTC)[reply]

Counts for "matrix is represented by" are misleading, because most (nearly all?) of the results are something other than what you are thinking of; stuff like "sparse matrix is represented by three arrays" or "epoxy matrix is represented by a non-linear viscoelastic constitutive model", "game payoff matrix is represented by the normal fuzzy numbers", "subdeterminant of a contingency matrix is represented by linear combination ...", "Each element of matrix is represented by ...", "The Lah matrix is represented by ⁠${\displaystyle L_{n}}$⁠", "hydraulic resistance of a clogging matrix is represented by an equation", etc. –jacobolus (t) 05:57, 20 June 2025 (UTC)[reply]

Hence the next line Sheer numbers on a snippet of text don't mean much taken out of context. – Farkle Griffen (talk) 05:59, 20 June 2025 (UTC)[reply]

"matrix is denoted by" gives results like "If the matrix is denoted by ⁠${\displaystyle M}$⁠", "The identity matrix is denoted by ⁠${\displaystyle I}$⁠", "The spectral norm of a matrix is denoted by ⁠${\displaystyle X}$⁠", and so on. –jacobolus (t) 05:59, 20 June 2025 (UTC)[reply]

If ⁠${\displaystyle A\in \mathbb {R} ^{3\times 4}}$⁠ then I would say ⁠${\displaystyle A}$⁠ is a "(3-by-4) table of (real) numbers". I guess if you want to get pedantic this may depend on what "⁠${\displaystyle \mathbb {R} ^{3\times 4}}$⁠" is defined to mean, and what "table" is defined to mean. But I don't think people are confused or mathematicians are likely to object if you say that an element of "⁠${\displaystyle \mathbb {R} ^{3\times 4}}$⁠" is a "table". –jacobolus (t) 23:32, 19 June 2025 (UTC)[reply]

Representation has particular meaning in mathematics and rectangular array can represent many things, not to say linear maps between finite-dimensional vector spaces. Fine with changing expressions to mathematical objects by the way.

Although doubly-indexed family or map of the form ${\displaystyle [m]\times [n]\to K}$ (or ${\displaystyle [m]\to K^{[n]}}$ or whatever) is not the same as rectangular array, they are interpretations or mathematical models for rectangular array. So it is fine, and very common, to say "a matrix is a rectangular array". 慈居 (talk) 11:55, 20 June 2025 (UTC)[reply]