Wikipedia:Articles for deletion/2 × 2 real matrices


 * The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review).  No further edits should be made to this page.

The result was delete.  Sandstein  21:47, 24 February 2021 (UTC)

2 × 2 real matrices

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See Talk:2 × 2 real matrices.

Summary:This article is written as an essay: it consists of a succession of assertions, often vague and using undefined concepts. There are many pieces of proof, but the results that are supposed to be proved are rarely stated. Thus, everything in this article is WP:OR or WP:original synthesis.

It is not clear that there is enough sourced matter for an article with this title.

Transforming the article into a redirect seems to be not a solution as the two possible targets (Matrix (mathematics) and Square matrix) have almost no content specifically related to this title. D.Lazard (talk) 10:20, 9 February 2021 (UTC)
 * Note: This discussion has been included in the list of Mathematics-related deletion discussions. D.Lazard (talk) 10:20, 9 February 2021 (UTC)


 * Delete. Some comments:
 * The article is full of topics and terminology not found in any of the standard linear algebra textbooks, such as split-quaternions, involutory matrix, split-complex numbers, and profile. It certainly seems like WP:original synthesis.
 * The purpose and level of the article are not clear.
 * There is an odd emphasis on equi-areal mapping (a top-level topic in this article).
 * Strangely, lowercase letters are used for matrices, except for the identity matrix.
 * It is not useful to say things like "Two matrices have a sum given by matrix addition".
 * The definition of matrix multiplication is not given in a precise way. (What does one do with the dot products? Which ones go where?)
 * There are many vague or otherwise strange statements, such as "M(2,R) is a union of planar cross sections that include a real line" and that for the split-quaternions "there is a similar union but with index sets that are hyperboloids".
 * Given all this, deleting the page seems warranted. It would be easier to rewrite the article from scratch than to try to fix all the issues, though it is not clear that there is a need for an article with this title. Ebony Jackson (talk) 19:06, 9 February 2021 (UTC)
 * If this article needs rewriting from scratch then that can be done by simple editing, without deleting it first, and I would have thought that it's pretty self-evident that this title should either be that of an article or a redirect. Most high-school students encounter two by two real matrices, but many do not encounter any other matrices. Phil Bridger (talk) 19:18, 9 February 2021 (UTC)
 * I'm not so sure about that; if you take an algebra class that uses matrices to solve systems of linear equations, you'll probably see 3 × 3 after you see 2 × 2, at least. XOR&#39;easter (talk) 14:23, 12 February 2021 (UTC)
 * I bow to your superior knowledge. It's well over 40 years since I was at high school, and I expect that things have changed since then. In my day two by two real matrices were used to illustrate linear transformations of the plane, but I don't remember having studied any matrices beyond that at the time. Phil Bridger (talk) 20:44, 12 February 2021 (UTC)


 * LinearFields.png Keep but rename and rewrite.
 * IMO there is important material here, and there is unity of content (ie it does all deserve to be taken together in one place), but this is absolutely not an introductory article about 2 x 2 matrices for people meeting them for the first time eg in the context of high-school mathematics, so IMO the current name of the article is inappropriate; and it also needs a much stronger lead to clarify exactly what it is about.
 * The real subject I think we have here is how the different possible linear transformations around a 2D fixed point can be classified into different broad groups, see eg Linear_dynamical_system or . (cf also the thumbnail right) This is a basic topic covered in introductory courses on dynamical systems to motivate the idea of different possible sorts of behaviour around fixed points; it is also useful background to have in mind whenever eigenvalues of systems are important, where the existence of either distinct real eigenvalues or alternatively a conjugate pair of complex eigenvalues can distinguish solutions into different qualitative sectors, with the possibility that small changes to the system may produce qualitatively interesting bifurcations between the two.
 * References and discussions of these different qualitative sectors of behaviour should be straightforward to find in books on nonlinear dynamics and bifurcations. As far as I remember from the '90s there was material on this in the introductory books on nonlinear dynamics by eg Thompson, Glendinning, or Arrowsmith. I am sure there would be similar material in other similar introductory books, as well as texts at a more advanced level on bifurcation theory.
 * Discussions in the dynamical systems materials tend to get to the different possible behaviours via considering the different possibilities for eigenvalues and Jordan Normal Forms. I think following that line here would be useful and would strengthen the article.
 * The article then specialises to the case of area-preserving transformations (also a topic of considerable interest in dynamical systems). The two generic sectors of transformations now correspond to rotations, which can be represented by a unit complex number, or to an area-preserving squeeze mapping (a.k.a. a hyperbolic rotation), which can be represented by a unit split-complex number (the hyperbolic analogue of a complex number).  It is very useful to get a sense of how these are the twins of each other, and to see how they fall out of the mathematics of 2d transformations.  This paves the way, for example, to appreciating Lorentz transformations as a hyperbolic analogue of rotations, and understanding how both can be represented in geometric algebra, generalising up from complex numbers and split-complex numbers. (See hypercomplex number for an overview of these objects).
 * Finally the article looks at how the two distinct sectors can be understood when 2x2 matrices are used to express other types of transformations, for example in projective geometry. I think the material here could be expanded and made more immediate, but it seems to me a very relevant thing to also cover.
 * So, in summary: I think that there is a topic here, that coheres as a whole, and is worth an article.  I think the material we have currently is worth keeping as a starting point for that article.  But I think the current title is not right and should be changed (eg perhaps "Classification of 2D linear transformations").  I think content should be added, as to how the existing material relates to eigenvalues and Jordan normal forms.  And I think the article needs much more orientation in the lead, that it's about how the overall algebra of 2D linear transformations can be classified into different sectors, and how these sectors relate to specific sub-algebras.  But as I think there is a real worthwhile topic here, and IMO the present text gets us at least part of the way to it, I think the !vote has to be keep. Jheald (talk) 15:29, 10 February 2021 (UTC)
 * This discussion must eventually be closed by an administrator, which probably will not be a specialist of this subject. So, such technical comments must normally be placed in the talk page of the article, and replaced by a summary focusing on the WP:deletion criteria. IMO, this long post can been summarized as "Keep for transforming it in another article with another title and another subject". This is definitively not the right way to create a new Wikipedia article.
 * More precisely, the suggested new title is "Classification of 2D linear transformations", that is the classification of the elements of the general linear group $$\operatorname{GL}(2, \mathbb R).$$ There are several such classifications in the literature, but it is not said which one is considered here. I suspect that this is another one that is intended, which is forbidded per WP:NOR. In any case, the article is not about $$\operatorname{GL}(2, \mathbb R),$$ but about the ring of 2×2 matrices, which contains this group as its group of units.
 * Also the post contains contains several assertions that are mathematically nonsensical, such as "2D fixed point" and the assertion that the section on area preserving transformations is relevant here. It is a standard fact, is stated in several WP articles, that linear maps preserving areas are exacly the linear maps of determinant one. So, a section introducing differential forms is definitely out of scope in an article that considers only linear maps. D.Lazard (talk) 10:02, 11 February 2021 (UTC)

If this article is so bad, why is it used in Japanese Wikipedia and in Russian Wikipedia? So the nominator would deprive English Wikipedia readers access to something to which Japanese and Russian students have access. What’s happening here? Rgdboer (talk) 05:35, 13 February 2021 (UTC)  Relisted to generate a more thorough discussion and clearer consensus.
 * Comment I just deleted a whole lot of text that was under-sourced, hard to follow and generally of niche interest (that 2-by-2 matrices can be written as split quaternions comes up far less often than wanting to find their eigenvalues, for example). XOR&#39;easter (talk) 19:10, 11 February 2021 (UTC)
 * My cleanup attempt was undone, so I restored the "reads like an essay" tag. At the moment, restricting to real matrices seems rather artificial to me. Yes, for some applications a real matrix is what you use, but even there you can end up talking about complex numbers anyway (e.g., eigenvalues for a dynamical system with circulation around a fixed point ). XOR&#39;easter (talk) 14:20, 12 February 2021 (UTC)
 * Delete The suggestions for fixing the article would seem to make it a page of Theseus. The title would have to be changed, the content rewritten, the focus altered, the lede changed to summarize the rewritten body — and, probably, the sources replaced with more standard ones. I generally prefer it when articles can be fixed rather than deleted, but the topic implied by the title is artificially narrow; writing better content about it would naturally end up writing about a broader class of mathematical objects. XOR&#39;easter (talk) 19:58, 12 February 2021 (UTC)
 * Keep
 * The Wikipedia projects in different languages have different standards for inclusion; whether an article exists in one doesn't really say much about whether it should exist in another. XOR&#39;easter (talk) 15:26, 13 February 2021 (UTC)
 * Delete. We do not have evidence for this as being a topic that is independently notable from quaternions and/or matrix operations in general. —David Eppstein (talk) 17:41, 16 February 2021 (UTC)

Please add new comments below this notice. Thanks, Natg 19 (talk) 05:49, 17 February 2021 (UTC)
 * Confirmed delete. (I am the opener of this thread, and, I have not explicitely said that my !vote is "delete", although it should be clear from my first post.) D.Lazard (talk) 16:48, 17 February 2021 (UTC)
 * Comment . The object of this article is the same of that of Split-quaternion must be kept, at least as they deal with isomorphic structures that have exactly the same properties. Thus merged them may seem a good idea (such a merge has been proposed, and resulted to "keep"), but it is not. Split-quaternion must be kept although the topic is rarely used in modern mathematics, at least for its historical interest. Split-quaternion was a mess, and I have almost completely rewritten it. In this rewrite, I have put more emphasis on the isomorphism of the two structures, but it appeared that redirecting 2 × 2 real matrices there is definitively not a good idea, as a reader searching information on 2 × 2 real matrices will certainly not find there what they are interested in. 2 × 2 matrix does not exist, and Real matrix is not an article but is a redirect to a definition in Matrix (mathematics). So there is no place for redirecting . As nobody has proposed any reasonable target for a redirect, and no specific encyclopedic content has been suggexted for 2 × 2 real matrices, I do not see any other solution than a deletion. D.Lazard (talk) 16:48, 17 February 2021 (UTC)
 * Confirmed keep: This article serves rotation matrix as a fuller explanation of imaginary units than given there, and links here. It also provides context for polar coordinates and a place for representation of split-complex and dual numbers. Thus there is a natural readership, indeed it averages 40 views per day. The nominator has not claimed any inaccuracy but has disparaged the references. He says it is OR but the topic arose first in the nineteenth century. Complaint that the content lies beyond elementary linear algebra texts is true because ring theory is encountered at an elementary level. A perturbing issue with the  nominator lies in use of row vectors in the projective  line section, as he has called this approach "fringe" at Talk:Real projective line. Reference to Row and column vectors shows several geometers preference  for row vectors when composing geometric transformations. As for the attitude to OR, see Talk:Split-quaternion where he is uninhibited. This article provides useful stepping stone for students moving on to abstract algebra or Lie theory. Rgdboer (talk) 03:56, 18 February 2021 (UTC)
 * It is true that rotation matrix had a section linking here. However the link was in a paragraph containing undefined notation, and the whole section used confusing terminology (for exemple the ambiguous term of "matrix plane" instead of "two dimensional subalgebra"). I have fixed this section, that presently does not links here.Also, "useful stepping stone" is a personal opinion that is not supported by any argument. Moreover WP:Wikipedia is not a textbook. D.Lazard (talk) 10:38, 18 February 2021 (UTC)
 * Certainly polar coordinates and rotation matrices are important topics, but the present article 2 × 2 real matrices does not provide a helpful way to learn about them, so one cannot say that they are "served" by this article. There is no linear algebra text, there is no abstract algebra text, and there is no Lie theory text that attempts to introduce the subject by presenting the selection of topics listed here. Ebony Jackson (talk) 20:09, 18 February 2021 (UTC)
 * D.Lazard was right to say that it is a fringe convention to write functions to the right of their arguments (I was arguing this too in my comments and references given at Talk:Real projective line), but in any case, that is a small point compared to the more serious issues with the article raised above. Ebony Jackson (talk) 20:40, 18 February 2021 (UTC)
 * Delete. Reads like unsalvageable incoherent mess, well into WP:TNT territory. Nsk92 (talk) 02:15, 19 February 2021 (UTC)
 * Delete. It's possible to write something about 2x2 matrices in particular (e.g. as 2D rotation matrices and representing complex numbers), but nothing of the current article contributes anything useful to the topic. Not even the section "2 × 2 real matrices as complex numbers" because it's confusing readers more than helping them. --mfb (talk) 18:46, 19 February 2021 (UTC)
 * Delete. This is an article without a clear purpose. The content of the article is a grab-bag of topics that use 2&times;2 real matrices but are not about 2&times;2 real matrices. Everything it says is better said elsewhere in individual articles on those topics. When that content is excluded from the present article, there is nothing left. Moreover, I can't see that ever changing. There is really nothing to say about 2&times;2 real matrices that could not be said of matrices generally, or all 2&times;2 matrices, or of the group SL(2,R), etc. On their own, 2&times;2 real matrices don't even have historical importance. Consequently I see no way that this article will ever contain useful content. Ozob (talk) 20:15, 19 February 2021 (UTC)


 * Comment. I agree that the article as written is a bit of a hodge podge of random topics that don't really belong on a page about 2x2 matrices, but I'm not sure I agree that 2x2 matrices don't deserve their own wikipedia page as a topic. Certainly there would be a great many people (most of whom are not mathematicians such as we are) who would go to wikipedia looking for a page that describe 2x2 matrices in an elementary way. These people would obviously be disatisfied with the page as it currently is, but I would think they would also be disatisfied if there was no page at all, and that they have to go to the examples in the general matrix page to find a discussion of 2x2 matrices that isn't general theory about nxn matrices. This may not align with the intended goals of the project, so I'm not presenting it as an argument to keep the page, but I think from the user perspective there would be demand for a page specifically about 2x2 matrices that present the nxn matrix theory in an introductory manner.Tazerenix (talk) 02:17, 20 February 2021 (UTC)


 * Delete. I thought that the description of this article might be hyperbole, and a cursory glance made it seem alright, but once I actually read each section it seems almost like a fever-dream, or some kind of mirror-version of a standard math article. The pageview statistics on this page make it seem like it's not a popular landing page for people googling, and the info people might want for 2x2 matrices seems like it would be better suited for a textbook than a wikipedia page. However, I believe someone could come up with a 2x2 matrix page, but it would be completely new content and essentially nothing from this page would belong in it, so I believe that this page should be deleted. Brirush (talk) 03:36, 20 February 2021 (UTC)


 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.