Wikipedia:Articles for deletion/Quaternionic matrix

Quaternionic matrix

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Unsourced with no indication of notability. Seems like fairly trivial OR to me: quaternions in a 2x2 matrix, and a Google search turns up nothing like this. JohnBlackburne^words_deeds 11:18, 31 August 2010 (UTC)[reply]

Keep: A trivial Google Scholar search returns tons of academic papers on the subject (the Google of the nominator must have been broken ). The information on the matrix multiplication is far from being trivial (at least for non-mathematicians). --Cyclopia ^talk 18:10, 31 August 2010 (UTC)[reply]
Please, show at least one link which deals with so named "octonionic product", either for quaternions or for matrices over another non-commutative ring. If there are no sources on such structure, then it must be removed and the rest of article become a trivial application of the matrix ring concept to the quaternions division ring. Incnis Mrsi (talk) 07:13, 2 September 2010 (UTC)[reply]
comment I did search, though a normal one not a scholar one, and turned up some of the same papers. But I could not find a definition, other than it's just a matrix of quaternions, or find anything that looked like this article (most of the articles seemed to be on more general n×n matrices). A search on both "Hamiltonian product" and "quaternionic matrix" turns up only mirrors of this page and a scholar search turns up nothing. So while the term exists much of the article seems unsourced. It would be good to hear from the page creator on this, as although as it stands it looks like OR it also looks like reasonable maths that could be sourced somewhere.--JohnBlackburne^words_deeds 19:07, 31 August 2010 (UTC)[reply]
- I don't understand where did you get the impression that the article is only on 2x2 matrices. --Cyclopia^talk 19:41, 1 September 2010 (UTC)[reply]
  - The definition ("matrix of quaternions") applies in any dimension and the first product given trivially generalises but the second doesn't, it only works as described in 2D. So in higher dimensions its not clear if this product is defined or if there are other maybe multiple products which simplify to this in 2D.--JohnBlackburne^words_deeds 21:05, 1 September 2010 (UTC)[reply]
    - Oh ok, I thought you referred to it being 2x2, not being bidimensional. --Cyclopia ^talk 22:48, 1 September 2010 (UTC)[reply]

- The definition is a bit of a tautology, and the sources were probably assuming readers would fill in the gaps. I'm thinking [1] probably covers the subject enough to establish notability. The article doesn't list any sources though, and it's unclear to me how much of the material should be removed. The definition alone isn't enough to justify the existence of the article.--RDBury (talk) 20:16, 31 August 2010 (UTC)[reply]
  - There is definitely material that can bring the article beyond a mere definition: e.g. here it is an article on eigenvalues of a quaternionic matrix. I am unfortunately far from having the mathematical competence to properly do that, I'm going to notify the appropriate Wikiproject. --Cyclopia ^talk 19:40, 1 September 2010 (UTC)[reply]
comment The determinant and therefore the eignenvalue problem are where you're going to have problems with such matrices. When calculating the determinant of a real or complex matrix you can do so in a variety of ways with order of multiplication usually ignored, but order matters when multiplying quaternions. A lot of the sources seem to be concerned with this problem, but more as something they're all trying to address, perhaps in different ways, rather than a known definition they can all agree on so which we can be pretty sure of.--JohnBlackburne^words_deeds 21:05, 1 September 2010 (UTC)[reply]
- Which is, if anything, a reason more to keep this article, not to delete it. It seems a lively topic in mathematics. --Cyclopia ^talk 22:48, 1 September 2010 (UTC)[reply]

comment We need definitions and results we can use for the article, which I don't see in any of the sources which all seem recent. It's not the place of Wikipedia to try and summarise ongoing, maybe even contentious, results. this looks more promising but I can't read beyond the first page.--JohnBlackburne^words_deeds 23:19, 1 September 2010 (UTC)[reply]

Well, no, Wikipedia is the correct place to report what RS (in this case, academic papers and books) report on a subject, regardless of its ongoing status and controversiality. --Cyclopia ^talk 23:34, 1 September 2010 (UTC)[reply]

Delete. Quaternionic matrices, i.e. square matrices whose elements are quaternions, are well-known and useful objects — for example, the compact symplectic group is realized as the maximal compact subgroup of GL(n,H). However, we already have an article matrix ring which deals with matrices over an arbitrary associative ring and I simply do not see the need for duplicating the same material in the special case of quaternions. On the other hand, the so-called "Hamiltonian" and "Octonionic" products, whose description at present comprises all of the article under discussion, are non-standard and possibly OR. (It is hard to be certain and it would be difficult to sort out notability issues because, historically, quaternionists have behaved as a sect, with their own societies and journals, and the fruits of their studies are quite isolated from mainstream mathematics.) Arcfrk (talk) 04:07, 2 September 2010 (UTC)[reply]

Thanks for the insightful comment. However I see in the literature many articles specifically dealing with quaternionic matrices, while the matrix ring article does not cite the special case of quaternionic matrices at all. It doesn't talk about eigenvalues, too, while there seems to be substantial literature about the problem of defining the eigenvalue of a quaternionic matrix. The article needs OR to be hashed out, but even if it is a special case, it seems a sourced one and therefore notable. --Cyclopia ^talk 07:53, 2 September 2010 (UTC)[reply]

Quaternionic matrices seem to have specific application in quantum mechanics that wouldn't be captured in a more generic article. Also, I don't think we should assume that everyone who might be interested in this is going to be familiar with more abstract ideas such as associated algebras.--RDBury (talk) 15:32, 2 September 2010 (UTC)[reply]

Comment The so-called "Hamiltonian" product is simply the usual associative product of matrix multiplication. If an n×n quaternionic matrix is represented using a 2n × 2n complex matrix or a 4n × 4n real matrix with particular structure, the product of the two quaternionic matrices on this definition properly matches up with the corresponding usual matrix product of the complex matrices or real matrices representing it. I am not sure about the meaning of a quaternionic determinant, and how it would relate to the corresponding determinants of the corresponding complex or the real matrices, but this is something that the article could usefully discuss. We have articles on various forms of matrices with particular types of structure, so I don't see an objection to an article on these matrices, if there is something interesting to say about them.

The motivation for the "Octonian" product is less apparent to me, and it could do with some sourcing. Possibly it is interesting and useful, but there seem to be no easily found references to it that jump out, at least at a surface level; and, perhaps I'm missing something, but I don't see an obvious connection between this product and the Octonian product of a pair of quaternions defined by the Cayley-Dickson construction. So this at the very least needs some clarification. Jheald (talk) 09:06, 2 September 2010 (UTC)[reply]

Weak Keep: For reasons given in my comments above. The article has severe problems and it might be better to start over from scratch, but I think some of the material is salvageable.--RDBury (talk) 15:32, 2 September 2010 (UTC)[reply]