Talk:Matrix (mathematics)/Archive 3

Introduction
It seems to me that the introduction is unnecessarily verbose. Some of the things (basic operations, applications, etc.), are already covered at length in later sections. It seems cluttered. I don't want to just start changing things, but this is something that we should look at. Lordcheeto (talk) 18:03, 25 January 2013 (UTC)
 * The introduction seems ok to me. An introduction should summarize the article. Rick Norwood (talk) 13:54, 26 January 2013 (UTC)
 * I don't think this is a verbose introduction. It is well structured, it contains only four paragraphs, it effectively summarizes an extremely complex topic. Paolo.dL (talk) 13:41, 14 May 2013 (UTC)

Why is taking a "submatrix" listed as a basic operation?
Matrix minors do not arise until we consider determinants or exterior algebra, and we need addition and multiplication of matrices before we can even think of those. (Co-, bi-)restriction doesn't arise until we've established the equivalence to the category of finite dimensional vector spaces, which requires multiplication, too. In this light, I think the secion on submatrices should be moved, or even removed and replaced with the discussion of matrix minors later in the article. &mdash; Kallikanzaridtalk 16:29, 11 May 2013 (UTC)


 * It seemed strange to me as well. However, you can explain submatrices without any reference to addition, multiplication and scalar multiplication. What about transposition and row operations? They are used in special contexts as well (e.g. solving linear systems). Paolo.dL (talk) 13:37, 14 May 2013 (UTC)

Rearranging the article
I think that the structure of the article has become somewhat messy, probably due to repeated minute edits and insertions by several different editors. It would be useful to rearrange it. It would become easier to browse, hence helping readers to extract the information they need.

Main kinds of matrices. It may be useful, for instance, to have a section listing the main kinds of matrices (row/column vectors, identity, diagonal, triangular, ... see list below). The "Definition" section currently includes subsections about "Row and column vectors", and "Identity matrix". They might be moved into a separate section (to be inserted after the "Operations" section) which will also include a subsection about "Square matrices" (now in a separate section), and other five subsections (see below). "Empty matrix" and "Infinite matrix" are generalizations, so the corresponding subsections can remain where they are (in a separate section about "Abstract algebraic aspects and generalizations").

Operations. The subsections about operations on square matrices (trace, determinant, eigenvalues and eigenvectors) may be moved into the "Operations" section (see below). Notice that "Decomposition" is currently in another section ("Computational aspects"). The lead of the "Operations" section will of course specify that Trace, Determinant and Eigen-values/vectors can only be applied to square matrices.

Proposed structure:
 * 1. Definition
 * 2. Notation
 * 3. Main kinds of matrices
 * Row vector or column vector
 * Square matrix
 * Identity matrix
 * Diagonal matrix
 * Triangular matrix
 * Symmetric or anti-symmetric matrix
 * Definite matrix
 * Orthogonal matrix
 * 4. Operations
 * Addition, scalar multiplication and transposition
 * Matrix multiplication
 * Other kinds of products
 * Row operations
 * Submatrix
 * Trace
 * Determinant
 * Eigenvalues and eigenvector
 * Decomposition.

Paolo.dL (talk) 16:23, 14 May 2013 (UTC)


 * Hmm.. I agree the article is currently feels disorganized, but I'm not convinced about having a large "operations" section. As Kallikanzarid pointed out, we should avoid doing complicated things too early; the current "basic operations" should be part of an introductory-type section, which lists only addition, multiplication, and scalar multiplication. Other things should be moved elsewhere and linked to their applications: like row operations to systems of equations, submatrices moved / replaced with matrix minors, etc. Mark M (talk) 08:40, 16 May 2013 (UTC)

Thank you for your feedback. I see your point. I won't touch the operations, but I had to rearrange a little bit the order and position of the subsections about "special kinds of matrices". For instance, "Identity matrix" is a special kind of square matrix, so I moved it into the section about square matrices. I also created a short subsection in the definition, listing the three main kinds of "size" (row vector, column vector, square matrix). Please feel free to move transposition, row operations, and submatrix elsewhere. Paolo.dL (talk) 17:41, 16 May 2013 (UTC)

Separate article about square matrix
Square matrix redirects to section "square matrix" in this article. This section is quite long and detailed. I propose to move it to a separate article, only shortly summarized here. There are special operations on square matrices (trace, determinant, eigenvalues and eigenvectors), and special kinds of square matrices (symmetric matrix, anti-symmetric matrix, Hermitian matrix, positive-definite matrix, orthogonal matrix). Any of these operations on square matrices or special kinds of square matrices has its own article. Similarly, square matrix should have its own article. In this article, a short summary about square matrices could be given in a specific subsection of the "Definition" section (see below). Paolo.dL (talk) 16:23, 14 May 2013 (UTC)


 * I'm not convinced this was the right move, so I've undone your edits.. it seems like an unnecessary duplication to me. I would think most things that are important enough to go in a "square matrix" article, also have a place in the "matrix" article. Mark M (talk) 08:40, 16 May 2013 (UTC)


 * I think I lean on the side of having the new article, moving the stuff over and just summarizing the main things in this article. Both articles will be big enough and important enough in their own right. Dmcq (talk) 10:10, 16 May 2013 (UTC)


 * Agreed with Paolo.dL and Dmcq. Square matrices have numerous properties relevant to applied mathematics which "rectangular" matrices do not: namely determinants, permanents, multiplicative inverses, representing symmetric and antisymmetric order-2 tensors, powers, exponentials and functions of matrices (?), etc.
 * Incidentally we have an article on 2 × 2 real matrices in case people haven't come across it. M&and;Ŝc2ħεИτlk 19:26, 16 May 2013 (UTC)


 * Thank you for your feedback. I am not sure whether to cut something in section Matrix (mathematics) or just add something more to the separate article Square matrix. Paolo.dL (talk) 21:45, 16 May 2013 (UTC)

Definition, notation, size
I do not agree about moving Notation and Size inside the Definition section. I would prefer to have three distinct sections: There's nothing wrong if we have a separate section in which we only define the term "matrix", its "elements", its "rows" and "columns" and nothing else. It is very short, but it makes the structure of the article simpler and easier to grasp. We do not need to define special kinds of matrices (e.g. square matrices), or explain notation there. I cannot see the reason why creating a big container such as "Definition and notation" can be useful to readers. Paolo.dL (talk) 16:40, 17 May 2013 (UTC)
 * 1) Definition
 * 2) Size
 * 3) Notation


 * From WP:BODY: Very short or very long sections and subsections in an article look cluttered and inhibit the flow of the prose. The sections you are proposing seem too short. It's helpful to readers because they can find out what the basic terminology means in one place: that first section. Mark M (talk) 07:54, 18 May 2013 (UTC)

Terminology is everywhere, even in "Basic operations". WP:BODY does not apply. We already have three "pieces" of text, and three headers. You propose a section with two subsections, I propose three sections. The only difference is that you give more importance to the first header. My proposal is neither more nor less "cluttered" than yours. However, Definition and Notation are two distinct topics, they do not need to stay in the same section. Indeed, your section title is "Definition and notation". Moreover, your solution is imperfect, as a container such as "Definition and notation" should include two subsections:
 * 1 Definition and notation
 * 1.1 Definition
 * 1.1.1 Size
 * 1.2 Notation

As you can see, since "Size" is a definition of some special kinds of matrices (row vectors, ecc.), "Size" should be a sub-sub-section within "Definition". This means that you need four headers (Definition and notation, Definition, Size, Notation). And this "looks cluttered and inhibits the flow of the prose" (see WP:BODY) even more than the three titles that I proposed above. There is a possible compromise between your approach and mine, that might help us to reach an agreement:
 * 1 Definition
 * 1.1 Size
 * 2 Notation

For instance, compare this with the similar structure of Euclidean vector, in which definition and notation are contained in separate sections (i.e. "Overview" and "Representations"). Paolo.dL (talk) 22:18, 19 May 2013 (UTC)

Rectangular
I'm still wondering what a rectangular array is. Nijdam (talk) 07:02, 26 August 2013 (UTC)
 * As far as I can remember, every textbook treatment of matrices that I have ever seen simply leaves the terms undefined and refers to a diagram. There are some things in logic and mathematics that are very difficult if not altogether impossible to define.  Outside of advanced treatises on the foundations of mathematics, most authors admit that they are being somewhat sloppy in some of their definitions in order to avoid being unnecessarily tedious and verbose. — Anita5192 (talk) 20:55, 26 August 2013 (UTC)

Dominant typeface for denoting matrix variables
This article predominantly uses uppercase roman (i.e. non-italic) bold typeface to denote matrix variables. I think that the standard convention is uppercase italic non-bold, even when vectors are denoted with lowercase roman bold. We should follow the dominant notation in the article. — Quondum 14:10, 30 September 2013 (UTC)

Principal submatrix
My textbooks do not agree on the definitions of submatrix or principal submatrix. Nor do web sites on the internet. If someone has a trustworthy source of the general definitions of submatrix and principal submatrix, please correct the definitions in this article, include the references, and cite the text. — Anita5192 (talk) 20:34, 1 December 2014 (UTC)


 * What is the definition in your textbook for a submatrix and principal submatrix? wpoely86 (talk) 14:59, 12 December 2014 (UTC)
 * What about:
 * http://www.tandfonline.com/doi/pdf/10.1080/03081089308818276
 * http://dx.doi.org/10.6028/jres.063B.003 — Preceding unsigned comment added by Wpoely86 (talk • contribs) 15:07, 12 December 2014 (UTC)


 * I just inserted two citations for submatrix. The only definition I can find in my books for principal submatrix is, "If A is a square matrix, then the principal submatrices of A are the submatrices formed from the first r rows and r columns of A for r = 1, 2, ..., n." in Anton (in the references), p.347.  Papers on the internet have various different definitions of principal submatrix. – Anita5192 (talk) 00:21, 13 December 2014 (UTC)


 * I removed the text for principal submatrix because nobody has supplied reliable citations from journals or textbooks, and sources on the internet do not agree upon the definition. Until someone supplies reliable documentation for this, it should not be in the article. — Anita5192 (talk) 20:39, 31 January 2015 (UTC)


 * I don't agree with simply remove the principal submatrix part. Try searching for it on Wikipedia and you will see it's being used in several places. It's not acceptable to use a term that's not explained. Either you have to go rewrite all those articles, or add the part about the principal submatrix back. The fact that there is not one uniform definition for it can be handled. We can add a couple of sentences that the definition varies from author to author. But simply not saying anything about it is not a good solution. wpoely86 (talk) 09:50, 10 February 2015 (UTC)


 * The principal submatrix is now back in the article. I reworded it more generally, since the sources do not agree. — Anita5192 (talk) 00:58, 11 February 2015 (UTC)


 * OK, I think it's much better now. Good job! wpoely86 (talk) 11:19, 18 February 2015 (UTC)

Matrix rows/columns notation
I was looking for a standard notation for matrix rows or columns, that I didn't find over here. Some forums like this one propose at least two notations that I find coherent:

$$\text{For } \mathbf A \in \mathbb{R}^{n \times m} \; : \;

\mathbf{a}_{i,*} = \mathbf{a}_{i\bullet} = \{ a_{i,j}, \forall j \in [\![ 1 \dotsc m ]\!] \} \; \text{and} \;

\mathbf{a}_{*,j} = \mathbf{a}_{\bullet j} = \{ a_{i,j}, \forall i \in [\![ 1 \dotsc n ]\!] \} $$

I did not feel confident enough to edit this vital article without asking, but I think this would deserve a subsection in the Notation part. I will do the edit if you guys agree.

— Le Zinzographe (talk) 10:28, 18 March 2016 (UTC)

Glossary
Do we really need a glossary section? Do we really need to redirect readers somewhere when they read the phrase, "off-diagonal element?" I don't think so. If readers understand what a diagonal is, will they not also understand what an off-diagonal element is? — Anita5192 (talk) 04:59, 8 June 2016 (UTC)
 * maybe there's a better solution to 'where' to link, but I like pages being able to have a link target for as many specific concepts/phrases/terms as possible (It was more about the redirect from 8 other places which can be redirected anywhere eventually). where there's a common concept there's a potential for discovery, parallels in ideas between use cases;   It's not just the definition, it's the whole link structure which feeds back into search and emerging AI techniques like thought-vectors, improving auto-translate ability etc (see my user page). Every label we give makes the system smarter. In time the section could grow. (There's other parts I was wanting to re-word to provide link-targets but didn't want to rock the boat too much on such a big page all at once). It could be put in an entirely separate page, "glossary of mathematics", "glossary of linear algebra".   I went with a separate section a I figured a whole new page would cause more controversy. Fmadd (talk) 06:53, 8 June 2016 (UTC)

Also Regarding Rows and Columns. Which is correct? The article or the accompanying graphic?
I'm not a mathematician, but it seems very intuitive that "rows" represent a horizontal concept and "columns" represent a vertical concept. The main article corroborates this intuition and clearly identifies the rows and columns to be horizontal and vertical respectively. The accompanying graphic however seemingly has the rows and columns reversed. This is either an error which needs to be corrected, or it is optional and conventional to label them either way, which sounds improbable to me, but if true needs to be explained, or I am simply missing something and am confused and there is no conflict between the article and the graphic. Someone in the know please fix or illuminate. Thx — Preceding unsigned comment added by 97.125.86.137 (talk) 05:49, 16 June 2016 (UTC)


 * Which graphic are you referring to? In which section? — Anita5192 (talk) 06:24, 16 June 2016 (UTC)


 * I think the poster is confused by the fact that the first graphic has arrows reflecting that the (horizontal) rows are arranged below each other (vertically) and the (vertical) columns are arranged next to each other (horizontally). JPD (talk) 07:08, 16 June 2016 (UTC)


 * I think that's a fair point; the lead image could probably be improved to make it more clear which are the "rows" and which are the "columns", since that is quite important terminology when discussing matrices. Looking at commons:Category:Matrix several other languages dropped the word "changes" from the diagram. But I don't think the arrow in the diagram is helping, since it could be misinterpreted as saying "here is a column" (hence the anon's confusion). Perhaps it would be better, for a lead image, to remove those arrows, and just have one row and one column highlighted and labelled as "first row", and "third column", or something like that. Mark MacD (talk) 16:36, 16 June 2016 (UTC)

What I Have Learned
I have read the article and now know no more about matrices than I did before I read it. In other words, I have learned nothing. The article is obfuscatory, opaque and generally poorly written. The authors deserve an "A" in math and an "F" in writing. — Preceding unsigned comment added by Chris319 (talk • contribs) 08:34, 30 June 2016 (UTC)

Please add brief explanation of third dimension.
I came here seeking to learn the terminology for referring to the third dimension, and after skimming through the article and spot-checking it to find certain words, was disappointed that I couldn't find the answer to my question: For a three-dimensional matrix, i.e. having elements we can refer to with (i,j,k) or (x,y,z), we say the subscripts refer to row, column, and ...? Is it 'depth'? 'slice'? 'stack'? What's the word? MathWorks appears to call it a 'page'. I see no discussion of this third dimension (I often want to use it for storing numbers according to spatial position), but only discussion of two-dimensional ("flat") matrices. Is there really no established word for dimensions beyond rows and columns? -- Newagelink (talk) 06:50, 5 October 2016 (UTC)
 * 1) If it has more than two-dimensions it is an array not a matrix (a matrix is a two dimensional rectangular array).
 * 2) I don't think there are well established names for different "types" of subarray. This would be insanely cumbersome for a 10-dimensional array, say. Instead people just call them "subarrays" (of a certain dimension).TR 08:04, 5 October 2016 (UTC)
 * Thank you for this clarification. Would it be correct to say, then, that a matrix is a special case of array as a square is a special case of rectangle? -- Newagelink (talk) 19:26, 30 October 2016 (UTC)
 * Possibly. A matrix in mathematics is a two-dimensional array endowed with specific operations and meaning.  For example, a matrix usually represents the coefficients of equations of transformation from a vector space to another (possibly different) vector space.  Hopefully the Wikipedia articles make these definitions clear.


 * Various nomenclature is in use for subarrays, for example, flats, hyperplanes, and slices, but I don’t think they are standard.—Anita5192 (talk) 20:48, 30 October 2016 (UTC)

Scalar matrix
I suggest removing the recently added section, Scalar matrix. Other than the identity matrix, I don’t believe such matrices occur frequently enough or are important enough in mathematics to warrant main type status. — Anita5192 (talk) 17:01, 26 February 2016 (UTC)


 * So is this you are thinking of that there weren't any Diagonal Matrix supposedly, so,- which can not be warrant enough to main type. If not remove this contradictory section please. Looks clumsy at top. —Dev A nand Sadasivamt@lk 22:16, 12 September 2016 (UTC)


 * Anita!#@!. Considering Lambda, it is crest and trough and by Spherical. Can you please undo which was reverted again and bring up the article in shape again. Anyone please don't give up any terrific and terrified of answers rather make it neat please. Regret on regression is shame. It's like no Ordinal you accept. Even you had bothered me on another article by revert. I am not understanding what averts or adverts so. —§Dev A nand Sadasivamt@lk 07:36, 16 January 2017 (UTC)

Section about infinite matrices
This part has no clear structure, it looks like a random collection of notes.

The part about linear maps is misleading. The argument about finite number of nonzero entries in column is wrong, because in infinite dimensional space, vectors generally are not finite linear combination of basis vectors. See for example article about orthonormal bases in Hilbert space. 195.113.27.90 (talk) 12:58, 6 April 2017 (UTC)

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