Talk:2 × 2 real matrices

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 Field: Algebra

x vs ×[edit]

The title of this article says

2 x 2

instead of

2 × 2.

It should get moved. Before I do that, a question arises: why is the title plural? WP:MOS says titles shouldn't be plural unless either the phrase is used only in the plural or the article is about the set of things rather than about the typical thing (e.g. Paul McCartney is a "former Beatle", but the article is The Beatles). And why the parentheses? Isn't "2 × 2 real matrix" good enough? Michael Hardy (talk) 03:48, 26 April 2009 (UTC)

Yes, the article is primarily about the set M(2,R), though the set is described in terms of properties of individual elements and their relations: First, how M(2,R) forms a number ring and a vector algebra which is isomorphic to the set of coquaternions. Second, the splaying of M(2,C) by various planes Pm according as mm is −1, zero, or +1. Third, the set of mappings contains the subset of equi-areal mappings. This subset, also described at SL(2,R), has detailed illustrations in this article with the subplanes Pm, so the set stuctures are at the heart of the article.Rgdboer (talk) 18:35, 27 April 2009 (UTC)
The bias toward alphabetic initial characters in a title made for the parentheses. With numeric initial characters accepted, there is no reason to keep the parentheses.
Another issue is the target readership, novice or expert. The growing mathematical literature in WP sometimes is advanced. For instance, this article leads right up to the topic of one-parameter groups. I have not provided the link since I doubt the one-parameter group article helps this one. The link to the special linear group, which is a well-developed article, will give a reader a suggestion of Lie theory. As for categories, the ring theory category was replaced with category:algebras. The article targets novice readers in ring theory that want to learn to delineate a commonly used ring.Rgdboer (talk) 22:44, 29 April 2009 (UTC)

Changing from row vectors to column vectors[edit]

I'm changing the article from using row vectors to using column vectors.

  • Our most mainline articles use Mxx' rather than xTMx'T, as at eg Transformation matrix, Linear map, Euclidean vector
  • I have also changed shear mapping, for the same reason.
  • Introductory textbooks also overwhelmingly introduce transformation matrices by their effect on column vectors, rather than row vectors.

A couple of reasons might be hypothesised for this:

  • It means the matrices L, M, N for successive transformations compose in the same order as is conventional for the corresponding group operators
  • It means the transformation Mx reads from left to right in the same order as an English sentence: verb first, then object.

For these reasons, it makes more sense for the exposition here to use column vectors, rather than the mathematically equivalent row vector forms. Jheald (talk) 08:19, 19 May 2009 (UTC)

See contrary opinion at Talk:Shear mapping. Response expected.Rgdboer (talk) 02:23, 20 May 2009 (UTC)

Profile[edit]

The notion of "profile" used in the article is not defined or explained. If two rings are ring-isomorphic, aren't their commutative subrings correspondingly ring-isomorphic? So in what sense do M(2,R) and the coquaternions have different profiles?  --Lambiam 10:51, 17 July 2009 (UTC)

The profiles are distinguished by the shapes of the sets of square roots of -1 and +1. In this article pertaining to M(2,C) these sets are hyperbolic paraboloids, as indicated. In the ring of coquaternions the set of square roots of +1 is a catenoid and the set of square roots of -1 is a hyperboloid of two sheets. Yes, the subrings are correspondingly isomorphic. The sense of the word profile as a cross-section corresponds to the isomorphic subrings; the sense of the word as a silhouette pertains to the index sets described that differ dramatically in their shapes.Rgdboer (talk) 22:00, 17 July 2009 (UTC)
Maybe this should go into the article because the main article on Ring (mathematics) never talks of "profile". Not being well-versed in ring theory, I had the same "huh?" moment as Lambiam when reading this page. 86.127.138.234 (talk) 05:42, 6 February 2015 (UTC)

Set of things[edit]

From User talk:Toby Bartels:

You have moved Real matrices (2 x 2) to 2x2 real matrix. If you had read the Talk page you would have seen that the MOS allows plural names when the subject is a set. The article has one section lower down that speaks of a typical matrix, but that is only after the full context has been set, that context being a set that displays three types of subrings, each associated with a type of complex plane. The dialogue with another editor on Talk had prevented this unfortunate Move. Please put the article back under the correct title or explain your reason for over-riding the MOS.Rgdboer (talk) 22:06, 27 August 2009 (UTC)

I don't see anything about this in the MOS; perhaps you mean Wikipedia:Naming conventions (plurals)? I don't think the exceptions there apply; that calls for using (for example) Legendre polynomials over Legendre polynomial, but that is still for an article about Legendre polynomials. If this article is really about the *set* of 2 × 2 real matrices, rather than about 2 × 2 real matrices themselves, then it should be at set of 2 × 2 real matrices or M(2,R). Even if the Legendre polynomials exception *did* apply, then the name should be 2 × 2 real matrices, not Real matrices (2 × 2); that looks like a disambiguated title, which it's not.

But I don't think that you have to fight with me; if you move this article to any of the following:

then I won't object, and I won't change it back. On the other hand, if you really insist on Real matrices (2 × 2), then please explain here.

Toby Bartels (talk) 01:23, 30 August 2009 (UTC)

We are not fighting, we are improving WP by consultation.
Of the three alternative you give, M(2,R) is the most direct, but too technical and unsuitable as a page heading in a simple matrix computation context like this article. Reviewing MOS we read about Nominal group (language) which is the object of the piped phrase "noun phrase". There the subject of modifier rank is discussed. In our case "real", as pre-modifier, is uppermost rank. The use of (2 x 2) as a post-modifier reflects a down rank of the matix order. Furthermore, in Wikipedia:Naming conventions (plurals) the Legendre polynomials case is raised as to illustrate "articles on groups of specific things". The present article clearly qualifies. It is not hard to anticipate an article on real matrices (3 x 3), in which case the post-modifier does the disambiguation as you anticipated.Rgdboer (talk) 21:51, 30 August 2009 (UTC)
I really don't know what page you mean when you cite "the MOS". But I don't think that you understand how disambiguation works. The only reason to use "(2 × 2)" as disambiguation is if the article might quite reasonably be at Real matrices, which is not the case. Since it is only about 2 × 2 real matrices, it should be at 2 × 2 real matrices or something like that. As for Wikipedia:Naming conventions (plurals), they are using "group" in a vague sense that's not synonymous with "set" in mathematics. That is, the article is about all of the individual Legendre polynomials at once, rather than about the set or sequence. (If you want to be technical, I'd argue that they're using "group" in the sense of plurality quantification —no article on that yet, I see.) So I think that this should be at 2 × 2 real matrix (if it's about the concept of a 2 × 2 real matrix) or at Set of 2 × 2 real matrices (or Algebra of 2 × 2 real matrices or something like M(2,R)) if its about M(2,R) as an object in its own right. All the same, if you really think that 2 × 2 real matrices is better, that's OK. —Toby Bartels (talk) 15:15, 31 August 2009 (UTC)

I've moved it to 2 × 2 real matrices. The article is about the set of all such matrices, along with the algebraic and topological structures that go along with that.

The original plural title was infelicitous in at least two ways, one of which is that it used the letter x instead of the × symbol. The singular title was problematic for the reasons already menntioned above. Michael Hardy (talk) 21:19, 31 August 2009 (UTC)

Read a matrix[edit]

If there are issues to discuss concerning the computations in the article section of this title, then they can be aired here. Until the issues are clarified, the section is being restored.Rgdboer (talk) 00:42, 8 January 2011 (UTC)

The problem is the total lack of references. I've been teaching Linear Algebra for thirty years, and have never run across this particular concept. This may be just a failing on my part, but I'd like a reference if you want this section to survive. Rick Norwood (talk) 18:24, 23 March 2011 (UTC)

The article is about an algebra over a field, a concept more involved than a vector space. In linear algebra the subject concerns what is true for dimensions of all magnitudes. Here a student learns everything there is to know about an element of a small algebra from hir knowledge of the whole. The algebra provides a description of all linear transformations of the plane, but only of the plane, not of space or the higher dimensions. Generally algebraists would not embarrass themselves with the trivialities of this case, so finding an explicit reference has been difficult so far. We can note that Elie Cartan evidently wanted to exclude it from his papers about transformations which "do not leave any plane invariant". Each non-singular 2 x 2 matrix does leave its complex plane invariant, so evidently Cartan was steering clear at that point. Students should appreciate that the classical groups lead to the study of Lie theory, a vast structure. Mathematics is such that readers might be challenged since authors know so much more, but don't provide complete details. The arithmetic can be checked. Remember that the time, when dimensions greater that three was preposterous, was only 150 years ago.Rgdboer (talk) 23:34, 23 March 2011 (UTC)

Karzel & Kist (1985) is the first reference, posted today. Expecting to find better. For algebraists this object frequently arises as "the other quaternions", as expressed in split-quaternions, where there is a good reference section.Rgdboer (talk) 20:49, 24 March 2011 (UTC)

My problem with this section is that nowhere in the article does it explain what is meant by "read a matrix". I am a first year undergraduate who is familiar with the basics of matrices, but I find this section and the article overall to be very unclear (compared to most of the math and science pages on Wikipedia). Can anyone expand this section to make it clearer to the non math graduate? 82.12.162.43 (talk) 08:55, 27 January 2012 (UTC)

I totally agree. I don't understand what "read a matrix" is supposed to mean. —Ben FrantzDale (talk) 16:46, 27 January 2012 (UTC)

Suppose z is a 2 x 2 real matrix that is not a multiple of the identity. Then z generates a subring of M(2,R). In the "Profile" section these subrings are described as planes in the 4-space that include a matrix m which squares to positive or negative identity or to zero. The exercise is posed at the outset: given z, find the Pm that contains z. Like reading off coordinates, one may read z as a point in the plane determined by an m basis.

I'm still not clear what "reading a matrix" means. I understand that powers of some 2×2 matrix z can be thought of as paths in R4. Is that really called "reading a matrix"? I've just never heard language quite like that. I've heard things like "can be thought of as" or "interpreted as" or "isomorphic to a point in R4".
Perhaps my confusion starts earlier in the article. The introduction makes sense. In the "Profile" section, I understand that the identity matrix scaled by a factor is isomorphic to the real number line. But "Since every matrix lies in a commutative subring of M(2, R) that includes this real line, the whole ring can be profiled by such subrings." doesn't make sense. The only other use of Profile like this I can find is Subring#Profile_by_commutative_subrings. (Profile doesn't include a link for any mathematical usage.) I think I'm partly thrown by the fact that this section does read like an exercise. Without motivation, it reads like an exercise, which isn't Wikipedia's writing style. Re-reading the conclusion of the section, it appears that the goal is to show that an arbitrary matrix z is isomorphic / can be thought of as / "can be read as" either a dual number, a complex number, or a split-complex number. It then notes that z can be "read" (interpreted? represented?) in polar coordinates. If that's what the section is trying to say, then I'd say we should rename the section "classification" or "interpretation" and open with "An arbitrary 2×2 matrix, z is isomorphic to an element of one of several rings." —Ben FrantzDale (talk) 14:14, 30 January 2012 (UTC)

Matrix as a complex number[edit]

Today the section title was changed to "Matrix as a complex number" for clarity. Additionally, an introduction was composed to note the undertaking. Thank you to interested readers for indicating necessary changes. The idea of powers of z illustrating the variety of planes is natural. Integer powers would be steps rather that a path. Certainly this idea of z behavior provides a motivation for wanting to know what kind of complex plane contains z.Rgdboer (talk) 01:36, 31 January 2012 (UTC)

Thank you Ben for making those changes today, including a more specific title to section.Rgdboer (talk) 01:31, 2 February 2012 (UTC)

References?[edit]

It is written above that this article is intended for the novice learning ring theory. Can someone cite a notable introductory textbook on abstract algebra or linear algebra from the last 20 years that discusses topics such as "profile of a ring" or "split-quaternions"? I had never heard of "profile of a ring" before, and after reading this article I still don't know what it means. And isn't "split-quaternions" an obsolete interpretation of the ring of 2 × 2 matrices? Ebony Jackson (talk) 21:22, 1 December 2013 (UTC)

This article is about an instance of a ring, not about ring theory. It does not advance a concept of "profile of a ring" but rather shows a structure like quaternion#H as a union of complex planes. Books on ring theory do not have space for particular instances, being dedicated to generalities. But M(2,R) does arise for instance, in F. Reese Harvey’s Spinors and Calibrations(1990) where there is a classification of Clifford algebras (see pp 208, 215, 223). The other face of M(2,R) as split-quaternions is the preferred form in the discussion of composition algebras. Being "obsolete" is an unlikely descriptor in mathematics.Rgdboer (talk) 02:14, 4 December 2013 (UTC)
Thank you for your response. Every notable introductory book I know covering ring theory (Artin, Jacobson, Lang, etc.) has plenty of examples of rings, including matrix rings, but none of these talk about split-quaternions. It would be best if the selection of topics of this page reflected what exists in introductory textbooks covering matrices and rings, if this article is intended for the novice. I have temporarily reduced the rating of the article to encourage people to fix this, and to provide references to document that the topics in this article are included in introductory textbooks. I also reduced the importance since it is debatable whether a separate article is needed for this topic given that matrix rings are already covered well by Wikipedia. It is almost as if we had a Wikipedia article on the cyclic group of order 5, another one on the cyclic group of order 6, etc. Ebony Jackson (talk) 01:39, 7 December 2013 (UTC)

Split-quaternion[edit]

KEEP:

Consensus here is not to merge. Furthermore, regarding the initial nomination, the guideline says that article b is a POV fork of article a only if it is "on the same subject" as a. The general notability guideline and the most relevant specific notability guideline I could find (on numbers) both indicate that this question of whether two articles are "on the same subject" is determined by the nature of the third-party sources the articles have, rather than by the mathematical relationship the articles' subject matters stand in. 2 × 2 real matrices and Split-quaternion have separate, non-overlapping lists of sources, thus suggesting that they are separate topics. It Is Me Here t / c 12:18, 1 August 2014 (UTC)

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Barely more than a PoV fork made by fans of hypercomplex numbers. These Hurwitz algebras are the same algebra. I hardly imagine a content that could be appropriate there but off-topical here. Incnis Mrsi (talk) 14:18, 11 February 2014 (UTC)

This would be the place to discuss a proposal to merge Split-quaternion into 2 × 2 real matrices, I presume?
  • Oppose. The concept of split quaternion has an independent history and that is enough for a spearate article. Deltahedron (talk) 15:55, 11 February 2014 (UTC)
    Which independent history, are you serious? The discovery by certain James Cockle in 1849? Or the use as one of numerous Macfarlane’s toys without a single reasonable application? Incnis Mrsi (talk) 16:14, 11 February 2014 (UTC)
Yes, that's right — I'm not in the habit of joking about these matters. As far as I can tell they were developed independently of matrices as an alternative to quaternions. Whether or not they turned out to be of use is part of the history. Deltahedron (talk) 16:59, 11 February 2014 (UTC)
I'll point out that they are of use in the theory of representations of the Lorentz group. Also, this argument is self-defeating; you're saying that these are the same algebra, but split quaternions don't have a "single reasonable application"... so 2x2 real matrices don't have a "single reasonable application", so why should they have a distinct page? MOBle (talk) 19:04, 24 February 2014 (UTC)
  • Oppose. In college studies, this article provides a bridge from linear algebra to abstract algebra. Furthermore, the content on equi-areal mapping, functions, and subrings is instructive for college students; placing these topics with Split-quaternion would be burial.Rgdboer (talk) 20:25, 11 February 2014 (UTC)
  • Support If I'm not mistaken, this, as a ring, is the same thing as split-quaternion, right? That's pretty decisive for me. -- Taku (talk) 01:27, 12 February 2014 (UTC)
  • Not yet. In my view there are other more important things to be done to both articles first. In particular, much more consideration should be given to WP:MTAA and WP:UPFRONT. The article apparently is envisioned by some as a gentle introduction to ring theory -- a university level subject. But its title is "2x2 matrices", a topic I was first schooled in at the age of 11. There's a lot that can be usefully said about 2x2 matrics before introducing ring theory, or even calculus.
So in my view it would be good to treat the geometrical properties first, using only the most elementary ideas -- eg first motivating in the simplest how each matrix identified with a transformation of a Euclidean 2d plane; explaining what it means for this transformation to be linear; motivate that the transformation can be split into an area-preserving element and an area scaling factor; deriving the determinant formula for the area scaling factor.
All of this can be done without talking about rings and subrings; definitely without talking about lines of matrices, something likely to be highly confusing to an eleven year-old only meeting the idea of co-ordinate transformations for the first time; and preferably using discussion in terms of unit vectors and unit area elements (unit bivectors), rather than infinitessimal differential forms.
Later one can then move on to how these elementary geometrical properties can be disussed at a more sophisticated level in terms of properties of rings and subrings. But that is not necessary to understand the basic geometrical analogy, so it should be left until later.
WRT the split-quaternion algebra(s), the key thing (to me) is that they are concrete examples of Clifford algebras -- Cl(1,1)(R) and Cl(2,0)(R) -- so they fit squarely into a system, and their geometric properties of eg being a means to encode rotations in 2+1 dimensions and 0+3 dimensions follow on directly from that.
It seems to me that only after the 2x2 matrix article has been heavily restructured with respect to its basics, should we be considering whether the split-quaternion material also fits in here. (I suspect a summary-style summary may turn out to be in order, but importation of the whole material would unbalance this article and be WP:UNDUE). But before we even think about that, this artice should be put in order first. Jheald (talk) 11:13, 12 February 2014 (UTC)
  • Oppose. For the same reasons as Deltahedron and Rgdboer. If quaternions are useful and distinct from other related concepts, then the same arguments lead us to conclude that split quaternions are too. "Isomorphic" does not mean "pedagogically equivalent". MOBle (talk) 19:04, 24 February 2014 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.