# Talk:Matrix (mathematics)

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)

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)
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)
That's definitely an improvement, thank you. Seattle Jörg (talk) 11:10, 28 July 2021 (UTC)
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 
  
    
      
        2
        ×
        2
      
    
    {\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)
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)
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)
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)
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)

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)

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

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)

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)
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)
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)
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)
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)
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)

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)

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)
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)
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)
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)
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)
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)
You are of course welcome to edit the article, or to make concrete proposals for changes.  --JBL (talk) 00:30, 9 October 2023 (UTC)
Yes, maybe I will. --Andres (talk) 00:48, 9 October 2023 (UTC)

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)
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)

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

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)

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)

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)


### Matrix (mathematics)
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, 
  
    
      
        A
        ∈
        
          
            R
          
          
            3
            ×
            4
          
        
      
    
    {\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)

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)
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)
I suppose it could be defined as a function 
  
    
      
        [
        1
        ,
        n
        ]
        ×
        [
        1
        ,
        m
        ]
        →
        
          R
        
      
    
    {\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)
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, 
  
    
      
        {
        1
        ,
        2
        ,
        .
        .
        .
        ,
        n
        }
        →
        S
      
    
    {\displaystyle \{1,2,...,n\}\to S}
  
, and matrices as a tuple of tuples (thus 
  
    
      
        {
        1
        ,
        2
        ,
        .
        .
        .
        ,
        m
        }
        →
        (
        {
        1
        ,
        2
        ,
        .
        .
        .
        ,
        n
        }
        →
        
          R
        
        )
      
    
    {\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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Okay, I'm pretty sure we agree on everything here, let me clarify what I'm saying.
I don't have an issue with "table" or "array" having an abstract meaning, the issue is they have both concrete and abstract meanings, and the former is, by far, more common. I was mostly upset with the previous version that said "a rectangular array or table of numbers, symbols, or expressions" which clearly defines a notation, not a mathematical object. I'm happier with the current version, but I still think it should be slightly clearer. Either my version above, or something like:
"... is a mathematical object defined as..."
The exact phrasing or emphasis on "Array" doesn't really matter to me, but I do think emphasizing "mathematical object" is important here. – Farkle Griffen (talk) 18:05, 26 June 2025 (UTC)
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)
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 ⁠
  
    
      
        
          L
          
            n
          
        
      
    
    {\displaystyle L_{n}}
  
⁠", "hydraulic resistance of a clogging matrix is represented by an equation", etc. –jacobolus (t) 05:57, 20 June 2025 (UTC)
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)
"matrix is denoted by" gives results like "If the matrix is denoted by ⁠
  
    
      
        M
      
    
    {\displaystyle M}
  
⁠", "The identity matrix is denoted by ⁠
  
    
      
        I
      
    
    {\displaystyle I}
  
⁠", "The spectral norm of a matrix is denoted by ⁠
  
    
      
        X
      
    
    {\displaystyle X}
  
⁠", and so on. –jacobolus (t) 05:59, 20 June 2025 (UTC)
If ⁠
  
    
      
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    {\displaystyle A\in \mathbb {R} ^{3\times 4}}
  
⁠ then I would say ⁠
  
    
      
        A
      
    
    {\displaystyle A}
  
⁠ is a "(3-by-4) table of (real) numbers". I guess if you want to get pedantic this may depend on what "⁠
  
    
      
        
          
            R
          
          
            3
            ×
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    {\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 "⁠
  
    
      
        
          
            R
          
          
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    {\displaystyle \mathbb {R} ^{3\times 4}}
  
⁠" is a "table". –jacobolus (t) 23:32, 19 June 2025 (UTC)
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 
  
    
      
        [
        m
        ]
        ×
        [
        n
        ]
        →
        K
      
    
    {\displaystyle [m]\times [n]\to K}
  
 (or 
  
    
      
        [
        m
        ]
        →
        
          K
          
            [
            n
            ]
          
        
      
    
    {\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)
@David Eppstein, can you explain your revert in a bit more detail, please? "Pedantic" isn't really a clear objection on its own, since it's essentially just saying "this is too correct", and clarifying "mathematical object" doesn't seem to me to be any more confusing than before. I am somewhat confused, though, since it seemed like you didn't oppose this phrasing before.
To the second edit, the current phrasing seems misleading, since it requires the reader click the link to undertsand that "represent" is not being used with its more common meaning, synonymous with "denote". Is there another phrasing you would prefer? (Though, to answer your edit summary, the verb phrase in the sentence is "[are] commonly related", but had a typo.) – Farkle Griffen (talk) 18:13, 30 June 2025 (UTC)
As stated directly above, 'it is fine, and very common, to say "a matrix is a rectangular array".' We don't need to add extra precision in the lead that confuses the naive reader by making the subject seem more technical than it is.
As for your no-verb non-sentence "Matrices commonly related to other mathematical objects by representation.": if you had written "Matrices are commonly..." it would at least be grammatical, but it does not help avoid the issues with the word representation (it is just a change of grammar and in both cases the reader can see that there is a link on the word), adds no actual meaning to the previous version of the sentence "Matrices commonly represent other mathematical objects.", and even after correction has unnecessarily complex grammar. —David Eppstein (talk) 18:31, 30 June 2025 (UTC)
I honestly don't see how "mathematical object" is any more confusing than before. It's not a particularly esoteric term and its meaning is pretty blatant.
To the second, I apologize for the typo "[are]". The phrasing was supposed to put "representation" as the emphasis in the sentence, rather than in the middle of a phrase, and the blue link could easily link to denotation. There is currently no way for the reader to know that "representation" does not have its more common meaning. If you diagree that my edit helps, that's fine. Is there another way to phrase it that would be clearer? – Farkle Griffen (talk) 18:47, 30 June 2025 (UTC)
It is false that the readers have no hint of a technical meaning of "representation". The hint is the existence of a wikilink on that word.
And mathematical object may not be particularly difficult, as technical terms of mathematics go, but in your version it is one more technical term of mathematics used to lard up the lead sentence when no lard is needed. Also your proposed lead's insistence that a matrix is only a matrix when it satisfies certain arithmetic rules is contrary to the body of our article, which states that those rules apply to certain matrices but also states that a matrix does not have to have numbers in it and does not have to obey those rules. —David Eppstein (talk) 19:15, 30 June 2025 (UTC)
And again, that link could easily be to denotation; common-language links are pretty common, e.g. the first sentence links the word "rectangular". The way that paragraph is written, "representation" does not have any emphasis in its sentence, and the rest of the paragraph can be perfectly well understood assuming "matrix" describes a notation. The reader has no reason to question the meaning of "representation". If you think adding emphasis isn't enough, then, again, is there another way to phrase it that would be clearer?
It isn't particularly technical nor exclusive to mathematics—It means almost exactly what any naive reader would assume it means. It makes the definition more technically correct at the expense of a few, easily-understandable, non-technical words. I don't think that's "lard". – Farkle Griffen (talk) 19:34, 30 June 2025 (UTC)
In response to your edit, currently § Basic operations is not mentioned in the lead at all, and I think you'd agree that it's a pretty important aspect of matrices. It could be rephrased to "...usually satisfying...". Is there another way to phrase it that you would prefer? – Farkle Griffen (talk) 19:40, 30 June 2025 (UTC)
Different authors disagree. Some claim that matrix entries need to be in some ring so that matrices support the expected arithmetic, while others let any objects sit in a 2d array and call it a “matrix”. I don’t think we need to belabor this though. –jacobolus (t) 19:52, 30 June 2025 (UTC)
I don't understand this reply. I'm agreeing it's not always necessary, but I'm saying it deserves a mention in the lead. Saying "usually" seems satisfying to me in mentioning it without going into detail. – Farkle Griffen (talk) 20:02, 30 June 2025 (UTC)

I have undone the addition of material about matrices over semirings and monoidal categories. It read like a very lengthy addition in comparison to how much material the article has on much less esoteric topics (like determinants and eigenvalues), as well as being written at a level that presumes the reader already understands the topic. (For example: "A 2-morphism between 1-morphisms 
  
    
      
        M
        ,
        N
        :
        A
        →
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    {\displaystyle M,N\colon A\to B}
  
 is a family of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
-morphisms 
  
    
      
        (
        
          f
          
            a
            b
          
        
        :
        M
        (
        a
        ,
        b
        )
        →
        N
        (
        a
        ,
        b
        )
        
          )
          
            (
            a
            ,
            b
            )
            ∈
            A
            ×
            B
          
        
      
    
    {\displaystyle (f_{ab}\colon M(a,b)\to N(a,b))_{(a,b)\in A\times B}}
  
. The definition of vertical and horizontal composition of 2-morphisms is natural.") This level of detail belongs somewhere, I'd say, but not here. Stepwise Continuous Dysfunction (talk) 03:04, 3 July 2025 (UTC)

If this material is to be used here or elsewhere, the "Matrices over semirings" reference has an error in the {{citation}} template ("날짜" should be "year"). Stepwise Continuous Dysfunction (talk) 03:16, 3 July 2025 (UTC)
Good day! An article should serve both as an introduction to beginners and as a reference for people who possibly already understands but wants to recall or find out more about the topic. I think I did enough in my edit to balance these two goals. For example, putting introductory words first, placing them in the generalization section, explaining less familiar notions, adding explanatory words to formal definitions, etc. Perhaps I can do more. :) In any case, deletion is not the best solution. 慈居 (talk) 03:31, 3 July 2025 (UTC)
I appreciate the effort, but I don't think the result as it stood was fully successful in that regard. As written, it would have fit better in the higher category theory ecosystem of articles (with perhaps a paragraph-sized summary here). Compared to the rest of the article, it's a big slab of incomprehensible to the bulk of the audience, and only of value to the narrow proportion of the readership who are already comfortable with esoterica. The fact that you had to say "The following further explanation requires familiarity with category theory" is proof enough that there was a problem. Stepwise Continuous Dysfunction (talk) 05:28, 3 July 2025 (UTC)
A problem? Sure. But I agree with 慈居 in that deletion is not the best solution. WP:Technical doesn't limit what kind of information is allowed in the article, only how it is presented. Either trimming or simplifying would be better, but outright deletion seems wrong. The content only needs to satisfy WP:ONEDOWN to meet GA standards, which, for this, is still fairly advanced. – Farkle Griffen (talk) 05:53, 3 July 2025 (UTC)
Matrix multiplication and matrix powering over semirings is definitely a useful thing, in tropical geometry and in shortest path and context free parsing algorithms, among other things. But I think that in such a basic article that covers so many bases, WP:DUE suggests that its coverage be limited to a sentence or two in Matrix (mathematics) § Abstract algebraic aspects and generalizations, not its own multi-paragraph stand-alone subsection. —David Eppstein (talk) 06:15, 3 July 2025 (UTC)
How much space it deserves, I'll leave up to other editors. I'm only opposing its outright deletion on the basis of "too technical". – Farkle Griffen (talk) 06:28, 3 July 2025 (UTC)
I think WP:DUE says quite the opposite. This article gives matrices with real and complex entries undue weight. It's just that it's acceptable per WP:MTAU. 慈居 (talk) 08:34, 3 July 2025 (UTC)
Quite the contrary, matrices with real or complex entries comprise the vast majority (99%? 99.9%? more?) of matrices used, especially in applications such as science or engineering, but even in mathematics. They should rightly be the primary focus of this article. –jacobolus (t) 15:24, 3 July 2025 (UTC)
I agree with jacobolus on this point. Stepwise Continuous Dysfunction (talk) 16:30, 3 July 2025 (UTC)
In mathematics matrices over fields or rings are the main focus. Real and complex matrices are just concrete examples. There are interesting special results for them (like Sylvester law of inertia for real matrices) but they do not apply to general matrices. The article wants real and complex matrices to be representatives of matrices in general. This is wrong, but acceptable to make articles less technical. 慈居 (talk) 22:52, 3 July 2025 (UTC)
The article doesn't "want" anything; it's just supposed to be representative of what's been written in reliable sources about the topic of matrices. That balance tips towards real and complex matrices, not further abstractions. Stepwise Continuous Dysfunction (talk) 04:57, 4 July 2025 (UTC)
Reliable sources are not limited to elementary linear algebra textbooks. There are hundreds and thousands of mathematical literatures treating matrices with more general entries than real and complex numbers. Real and complex matrices are just a tip of an iceberg. 慈居 (talk) 05:11, 4 July 2025 (UTC)
And there are tens of thousands of physics papers in quantum mechanics alone that use only matrices over real and complex numbers. The whole "iceberg" does not fit into one article. Stepwise Continuous Dysfunction (talk) 05:14, 4 July 2025 (UTC)
Matrices with general entries is also used in quantum mechanics, as in the quantum Yang-Baxter equation. Real and complex matrices are typical examples; still, there are many others. 慈居 (talk) 05:23, 4 July 2025 (UTC)
The existence of the odd application here or there of matrices with more general entries doesn't change the fact that real and complex matrices are by far the predominant ideas in, I dare say, every subject that applies linear algebra, from computer graphics to quantum statistical physics. Stepwise Continuous Dysfunction (talk) 16:40, 6 July 2025 (UTC)
At least, WP:DUE is a wrong policy to quote here. Neither of the applications of real and complex matrices you mention supports the idea that there are no other kinds of matrices. Many of them refers to them as real/complex matrices, and use notations like 
  
    
      
        
          M
          
            n
          
        
        (
        
          C
        
        )
      
    
    {\displaystyle M_{n}(\mathbb {C} )}
  
 which implies that there can be other rings of entries that can take its place. 慈居 (talk) 01:25, 7 July 2025 (UTC)
Being policy, WP:DUE applies everywhere.
The point is not that "there are no other kinds of matrices". It's that if most books do not talk about other kinds of matrices, then this article probably shouldn't say much about them either. When a book uses a notation like 
  
    
      
        
          M
          
            n
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle M_{n}(\mathbb {R} )}
  
, or 
  
    
      
        SL
        ⁡
        (
        n
        ,
        
          R
        
        )
      
    
    {\displaystyle \operatorname {SL} (n,\mathbb {R} )}
  
, it's not explicitly talking about matrices over other sets. A vague implication that one might read into the notation is not justification for anything. Stepwise Continuous Dysfunction (talk) 03:56, 7 July 2025 (UTC)
Oh yes, it does apply everywhere, it just does not support your point. The typical example of undue weight given in the policy, is the flat Earth belief, a conspiracy theory. The references I have provided are reliable references published in authoritative math journals. It proves the significance of the topic and that should be all. 慈居 (talk) 04:26, 7 July 2025 (UTC)
My point is that it is a very strong assertion to say that "other matrices are not significant". There are many (though not more) references that discusses general matrices, so a reference should be provided that claims that they are not significant, to support your idea. 慈居 (talk) 04:32, 7 July 2025 (UTC)
In some rare cases, this does happen. Like some though-to-be generalizations of metric spaces (e.g., rectangular metric spaces, if I remember correctly) is actually not a generalization, as shown in a published paper and cited in many. 慈居 (talk) 04:35, 7 July 2025 (UTC)
"A sentence or two" in the section about abstract algebra sounds appropriate. Even a short paragraph, along the lines of the one that currently introduces § Matrices with entries in a category, could potentially work. I agree that "its own multi-paragraph stand-alone subsection" is going too far. Dumping coproducts, bicategories, and coherent isomorphisms into an article read by high-school students is the kind of thing that gives Wikipedia its reputation for obscurantist, self-indulgent math content. Stepwise Continuous Dysfunction (talk) 16:43, 3 July 2025 (UTC)
I think the section title (Abstract algebraic aspects and generalizations) justifies technicality of my edits. There are other technical materials there also more than a sentence or two (like Matrix groups).
Also, WP:TECH-CONTENT suggests not to remove on-topic technical contents. High-school students are not the only readers. Maybe they are for now, since the article is not so useful to others, but I believe that should change. 慈居 (talk) 23:07, 3 July 2025 (UTC)
That guideline is entitled, "Make technical articles understandable". A lengthy excursus on category theory, written in a way that presumes subject-matter expertise, does not make the article more understandable. The trouble is that a lengthy and detailed treatment of categories and semirings is not on-topic for an article that needs to give an overview of the subject of matrices. Stepwise Continuous Dysfunction (talk) 05:01, 4 July 2025 (UTC)
The guideline tells what to do and what not to to "make technical artices understandable". It tells not to remove on-topic contents just because they are technical. It tells not to oversimplify.
They are called matrices over semirings and the bicategory of matrices. Please see the reference I have provided in my edits. They are on-topic by all means. If you don't think so, please provide a reference as I did. 慈居 (talk) 05:15, 4 July 2025 (UTC)
The fact that some subjects are intrinsically technical means that those subjects are intrinsically ill-suited to some articles. The page Quantum mechanics is the wrong place to try and explain, for example, Lieb and Ruskai's proof that von Neumann entropy is strongly subadditive. Quadratic formula would be the wrong place to try teaching Galois theory, and even more so for a section that can only be understood by readers who already know how category theorists think of Galois theory.
It's not the lack of a reference that made the material off-topic. It's the fact that this is, by its nature, the most introductory article on matrices that Wikipedia has to offer. Stepwise Continuous Dysfunction (talk) 16:37, 6 July 2025 (UTC)
Your description of on-topic-ness goes against WP:TECH-CONTENT which describes on-topic-ness as relativeness of the contents, not technicality.
The proof certainly should go to von Neumann entropy since it already exists, not because it'd be off-topic. My edits, although many thinks that it is detailed, is only a brief summmary and not enough for an article separation.
Quadratic formula should not mention Galois theory because it does not need that. Cubic equation discusses the Galois group of the cubic polynomial, although some readers may not be familiar with Galois theory. It's not about teaching Galois theory. 慈居 (talk) 01:54, 7 July 2025 (UTC)
WP:TECH-CONTENT says, "When trying to decide how much technical detail to include, it may be helpful to compare with a standard reference work in the particular technical field." What standard reference work about linear algebra brings in higher category theory? Since many reference works about linear algebra exist, how many of them discuss the bicategory of matrices versus, e.g., properties of the determinant? On what grounds can we say that this, out of all the true things that have been written about matrices, is so important that it has to be included here?
Moreover, pointing at a guideline behind the scenes doesn't actually make the text more comprehensible. Your addition achieves brevity by assuming that the reader already knows the meaning of "coproduct", "initial object", "1-morphism", etc. In other words, it manages to be short by not caring whether it can be understood. This is out of place in an article which, just a few paragraphs up, takes the time to define what a group is. The upward shift in the level of presumed background knowledge is jarring. And just saying "don't expect to understand this if you're not already among the elect" is hardly a solution. Stepwise Continuous Dysfunction (talk) 03:35, 7 July 2025 (UTC)
Elementary lInear algebra is not all of the linear algebra. Li Wenwei, Algebra methods. Volume II, linear algebra (代数学方法.第二卷,线性代数) discusses abelian categories, complexes, triangular categories, spectral sequences, etc. which, to quote the preface of the reference, "fairly are topics of linear algebra".
They don't need to understand what coproducts are. They only have to roughly understand that there are generalization of matrices where something called coproducts has role of ring addition, something called tensor product has role of ring multiplication. "It would be a wrong place here to teach general category theory here." 慈居 (talk) 04:23, 7 July 2025 (UTC)
Quadratic formula has a section Quadratic formula § By Lagrange resolvents which should at least point interested readers in a useful direction. I think it could be okay to have various even more technical sections included there if anyone thinks it would be valuable to plausible readers of the article and there's enough to say specifically about the relation to the quadratic formula per se. Just linking out for more detailed coverage is probably better though if it's going to be significantly more general and abstracted. –jacobolus (t) 02:55, 7 July 2025 (UTC)
Wouldn't matrices with other than real and complex entries be a natural break point to split off an article on Matrix generalizations? Matrices with real and complex entries are so important in so many applications, and so challenging to non-mathematicians, they should be enough for an article. The generalizations article could then be as technical as we wish and, of course category theory aspects would be fully appropriate.--agr (talk) 10:47, 3 July 2025 (UTC)
Yes, that would be a natural break point to split off an article about generalizations, leaving a concise summary here per usual Wikipedia style. Stepwise Continuous Dysfunction (talk) 16:38, 6 July 2025 (UTC)
The amount of material currently in § Abstract algebraic aspects and generalizations seems fine. I wouldn't try to make any shorter a summary than this; indeed the section could plausibly be moderately expanded in place. Expanding further in a separate article or articles (without removing much from here) would be fine though. –jacobolus (t) 23:53, 6 July 2025 (UTC)
I think it's helpful to introduce the relevant paragraph in the lead section by saying that matrices commonly represent other kinds of objects (this includes especially linear transformations). We discuss this in more detail and variety later down the page. "matrices are used as linear maps" also seems confusing, comparable to saying "decimal numbers are used as temperatures". –jacobolus (t) 15:29, 3 July 2025 (UTC)
I agree if the sentence does not link representation (mathematics). 慈居 (talk) 23:18, 3 July 2025 (UTC)
Did you read that article? It says:
a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yi conform, in some consistent way, to those existing among the corresponding represented objects xi.
Under this definition it is entirely reasonable to say that matrices are often used to "represent" other kinds of objects (or I guess if you want to be pedantic you might say that collections of matrices are used to represent collections of other objects). –jacobolus (t) 05:31, 4 July 2025 (UTC)
Ah, I thought that article is about representation theory. But do we really need a link for the word "represent", apparently used in a literal sense? 慈居 (talk) 05:35, 4 July 2025 (UTC)
To be clear I'm completely fine with saying "matrices represent many objects"; I just don't want the link. 慈居 (talk) 05:38, 4 July 2025 (UTC)
Without a wikilink would be fine; I included one to make it more explicit that it's not only about representation theory. –jacobolus (t) 23:54, 6 July 2025 (UTC)