Talk:Matrix (mathematics)

Coefficient of Linearity

When I search this term,- Coefficient of Linearity as a general search in Google, Lucky link is Coefficient of determination, I mean the first one what I am getting to.

But is that suffices, I have eye sight problem, unfortunately, Yes I read determination as differentiation at first sight. So got to writ over here again. I hope you folks can feel the difference. Maybe there is one, say Nabla or so.

First when I look at the article, it needs to be written even more as because the term Modern Algebra not referred. Means it can be driven with however the way you calculate so, addition, subtraction and so on of Matrices. Considering the one or more stands as Principal. Because it was not clearly explained by anyone so, rather been talked like what your book say so about it in this Talk page,- Talk:Matrix (mathematics) earlier & concluded and been edited, accepted.

Polynomial expansion clearly can be segregated for sure to more than one Linear Equation at its granularity, unless it reportedly dual by nature which might occur probably in Binomial Expansion very commonly. In any given non-linear equation, there exists at-least one Linearity. For anymore complexity, maybe by Principal,- Principal Matrix. So Editorial Team may intervene.

I disagree with Matrix addition, subtraction even if it is just Mathematics.

All I need is reference of Modern Algebra. It can be written well to have this compliance for fulfillment maybe to have clarity on Mathematical Proofing,- right, like, this has to be exactly should be done like this or could be done like this. The clarity that this Article needs.

—Dev Anand Sadasivam^t@lk 18:48, 11 August 2018 (UTC)

History

The section on 'History' mentions a very early use of matrices in China, then says that Cardano 'brought the method to Europe' in the 16th century. This might be interpreted as meaning that Cardano was aware of the ancient Chinese example and then introduced it to Europe. This seems highly unlikely. If it is not the intended meaning, I suggest the text should say just that Cardano was the first mathematician to use the method in Europe. Incidentally, the article on Cardano does not seem to mention his contribution to the subect.109.150.6.195 (talk) 20:33, 29 December 2019 (UTC)

Done. Seattle Jörg (talk) 07:40, 27 July 2021 (UTC)

See also section

I have edited the "See also" section for displaying the short descriptions of the linked articles. I leave to others to decide which links are relevant here. D.Lazard (talk) 11:29, 3 May 2020 (UTC)

Rectangular matrices with exact Inverse

Years ago, I seem to recall a Wikipedia page that showed examples of rectangular matrices that did not require SVD. I think some carried names. I've searched Wikipedia and Google, and now I find nothing. Any ideas on where to find such examples? Charles Juvon (talk) 21:25, 3 September 2020 (UTC)

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)

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 $2\times 2$) 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)