Talk:Matrix (mathematics)/Archive 2

Archive 1

Archive 2

Archive 3

A better introduction than "rectangular array of numbers"

Does anyone have a good idea what to write in the introduction? In higher level mathematics, and mostly in system theory, one of the most important things is that a matrix should not be viewed as an array of numbers, instead numbers should be viewed as 1*1 matrices. --131.188.3.21 (talk) 16:54, 15 March 2009 (UTC)

I don't think this is a good explanation for somebody who does not yet know matrices. If you define numbers via matrices, how do you define matrices then (without using numbers)? Jakob.scholbach (talk) 07:23, 16 March 2009 (UTC)

You can define them, for example, as a collection of vectors, or as a description of a transformation. But you're right, the current introduction is the best for people unfamiliar with matrices. --131.188.3.21 (talk) 19:20, 16 March 2009 (UTC)

My suggestion would be "rectangular array of numbers or expressions", although I'm not sure that expressions is the right term - the elements of a matrix can (and often are) anything from numbers to other matrices. --Kragen2uk (talk) 05:30, 18 January 2011 (UTC)

This seems to be a good point, but maybe too fine a point for the lede. In knot theory, the matrices I work with have polynomial entries. That should be in the article somewhere. Rick Norwood (talk) 13:11, 18 January 2011 (UTC)

Notation

Can anybody confirm this? Jakob.scholbach (talk) 17:04, 27 March 2009 (UTC)

An alternate convention is to annotate matrices with their dimensions in small type underneath the symbol, for example, ${\underset {m\times n}{A}}$ for an m-by-n matrix.

I use that notation when I teach Linear Algebra, when I introduce matrix multiplication, but it isn't in the textbook I use. Rick Norwood (talk) 13:14, 18 January 2011 (UTC)

Entry or element

In all and every (printed) book or article on matrices in English I've seen, the thingos at given positions in a matrix are called "entries". The same is true for this article, but without defining "entry" explicitly. I've heard rumours of the usage of "elements" instead; and I notice that this usage - inconsistently - is employed in some other articles in Category:Matrix theory.

E.g., in the beginning of the section Matrix#Matrix multiplication, linear equations and linear transformations the ordinary definition of the entries of a product matrix is given. The section, however, refers to matrix product as its main article; and there the same rule is given for the elements of the product matrix, in the section matrix multiplication#Ordinary product. Actually, at a glance, I found no direct mention of "entries" anywhere in that article. However, there was an indirect reference, which actually might be a bit confusing for non-expert readers. In the section matrix product#Hadamard product, the concept entrywise product is mentioned, but again as defined by a multiplication of "elements".

Moreover, there seems to be no way for a non-expert to find a reference to matrix entries; at least, I found none. The page Entry is a redirect to Entrance, which is a disambiguation page. This page does not mention "entry" in any usage; not even the accountant's entries are mentioned. (Actually, I think that the accountant terminology is the historical reason for the term entry in connection with matrices.)

What I wonder is, first, is actually "elements" nowadays used in lieu of "entries" in some text books or articles in English, at lower or higher level; and second, is there any article on matrix entries anywhere in en:wiki? JoergenB (talk) 21:53, 5 November 2009 (UTC)

Actually, looking in Shilov's Introductiong to the theory of linear spaces for matrices as tensors, I found that here the entries indeed are called "elements". Now, it's about 40 years since I had Shilov as a text book; but I have looked in it since then, and ought to have remembered.

So, both "entry" and "element" do be represented in English text-books. Which term is nowadays most common? JoergenB (talk) 22:33, 6 November 2009 (UTC)

Element is overwhelmingly most common -- see below. Michael P. Barnett (talk) 19:24, 30 April 2011 (UTC)

Interpretation as a parallelogram

This section has a few errors and some mixed-up ordering that make it quite confusing.

First of all, at different places in the section, different sets of vertices are given for the parallelogram, some with (a,b) and some with (a,c) as a vertex. The picture suggests that (a,b) should be the right one.

Secondly, it isn't possible to multiply Ax if the x's are row matrices, since the number of rows in A (i.e. 2) does not equal the number of columns in x (i.e. 1). As written, Ax is not a valid operation, only xA is.

It would also help to clarify things if the transpose operations were kept separate from the multiplication operations. It makes the article unnecessarily confusing to have them interspersed.

I believe what the article should say is:

If A is a 2×2 matrix

$A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}\,$

then the matrix A can be viewed as the transform of the unit square into a parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d).

The parallelogram in the figure is obtained by multiplying each of the row vectors ${\begin{bmatrix}0&0\end{bmatrix}},{\begin{bmatrix}1&0\end{bmatrix}},{\begin{bmatrix}1&1\end{bmatrix}}$ and ${\begin{bmatrix}0&1\end{bmatrix}}$ with matrix A (which stores the co-ordinates of our parallelogram) in turn (i.e. xA). This gives us the row vertices [0 0], [a b],[c d] & [a+c b+d].

Note that since $xA=(A^{T}x^{T})^{T}$ ,we could equally have transposed A to $A^{T}={\begin{bmatrix}a&c\\b&d\end{bmatrix}}\,$ and x to ${\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}a\\c\end{bmatrix}},{\begin{bmatrix}a+b\\c+d\end{bmatrix}}$ and ${\begin{bmatrix}b\\d\end{bmatrix}}$ and calculated $A^{T}x^{T}$, getting the same set of vertices that xA returned, but represented as columns instead of as rows, which can be easily transposed back into the row matrices equal to xA.

13.1.100.136 (talk) 19:38, 3 February 2011 (UTC) Lottie Price

That's right. Not only did this section contain some inaccuracies, it also did not fit well into the article's section ordering. I merged (and corrected) its content with other sections. Jakob.scholbach (talk) 19:59, 3 February 2011 (UTC)

Change to multiplication

I have added parentheses to the example calculation to what I think is clearer, since I had to spend some time trying to disentangle the dots and the plusses. I also think that (now):

(the underlined entry 1 in the product is calculated as the product (1 · 1) + (0 · 1) + (2 · 0) = 1):

would be better as:

(the underlined entry 1 in the product is calculated as the sum (1 x 1) + (0 x 1) + (2 x 0) = 1).

But I didn't want to go so far. Myrvin (talk) 14:12, 3 April 2011 (UTC)

Spurious history

Maybe the date should be 1848, but even then I doubt it:

The term "matrix" is the Latin word for "womb" and was first used in mathematics by James Joseph Sylvester in 1948. He used this translation because he viewed a matrix as a generator of determinants.

BillWvbailey (talk) 14:30, 10 April 2011 (UTC)

The sources says 1848 or 1850. It was a typo. Supported by a number of RS: [1], [2], [3], [4], [5], and [6] p. 190. Please do not rv without RS stating this is an urban legend or something. Mhym (talk) 17:55, 10 April 2011 (UTC)

Gimme a break. As typed into the article this was utter crap and deserved to be reverted. Whether or not, after the date correction, it's still true is a matter for the historians. Bill Wvbailey (talk) 02:15, 11 April 2011 (UTC)

Wvbailey would do better to use polite language. But as for the matter of Latin, I must agree that there seems to be no justification to the claim that "matrix" is a Latin word for "womb", however many math books say so (probably the authors are copying one another here). See [[7]] which gives alvus, gremium, uterus, venter, and volva, but not matrix. In fact matrix does not seem to be a Latin word at all; it seems more like retro-fitted to a Latin form. The French word "matrice" (in the sense of "mold") is quite ancient, goes back to at least 17th century, possibly 13th century; my guess would be this served as model. Marc van Leeuwen (talk) 09:25, 11 April 2011 (UTC)

You're correct, I should have used the more scholarly word Bullshit. Actually when I saw the original entry with the bad date I was going to delete it as BS dropped on the page by a vandal, but as it was entered by a registered wikipedian I gave it the benefit of my extreme doubt and resorted to WP:BRD. My Webster's Ninth Collegiate Dictionary (1990) gives the origin of matrix as you guessed it: fr L. womb, fr. matr, mater [mother]. But I concur with your suspicions about the word; Webster's also gives as a main usage 2a as in "mold" (die, e.g. used for casting), so perhaps the (square) matrix casts out a determinant, perhaps not. Unfortunately we should resort to the Oxford Dictionary of the English Language (not available on line unless you subscribe for about $500 per year). My skepticism is more related to the claim of "first usage", which as you point out may be inherited lack-of-"wisdom". The claim may be true, but only deep academic-grade research into the first usage would reveal this together with a history of the notion of a "5a: a rectangular array of mathematical elements [etc]". Surely such things were called something but what was it? In my working on wikipedia articles I've gone through a number of these and am involved in one now, i.e. researching back to the first usage (if possible). Almost always the results are surprising and troubling. Almost always these involve a trip to an academic library and hours (even days) of research. Only in this way can we be sure that we're not involved in passing on bullshit. Bill Wvbailey (talk) 14:37, 11 April 2011 (UTC)

Here is what OED says in the entry for matrix :

Etymology: < classical Latin mātrīc-, mātrīx, female animal kept for breeding, in post-classical Latin also (from early 3rd cent.) womb, source, origin, apparently < māter mother (see mater n.1), with alteration of ending to -trīx-trix suffix, perhaps after nūtrīx wet-nurse, nurse (see nutrix n.).

It also lists couple of dozen examples. Here is the oldest quote OED is using in the "mathematics" section, which actually supports the historical claim:

(1850) J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 369 "We‥commence‥with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order."

I say there is a clear need here acquiesce to authority, rather than try OR on this. I will restore the original quote as without it being present on the page the discussion is moot. Mhym (talk) 17:31, 11 April 2011 (UTC)

Your quote from the OED supports the first half elegantly. But any inference to "wombs" is not supported by the quote, nor its context. Here's why: The word, at least in my Websters has the usage 2a. 2b. the natural material in which a fossil [etc] is embedded, material in which something is enclosed. I know in engineering a "matrix" can be used to describe a substrate, e.g. a woven cloth of a circuit board. And considering the Elizabethan Victorian/Edwardian interest in paleontology I, Bill, would surmise that the word "substrate" is the intended meaning (synonym), not "issuing from a womb". But this is just my supposition, I cannot support it by fact or hard evidence. If "hard evidence" is available, e.g. one of your sources above states that this womb-business is the case, then you can put that reference in together with the supposition, but only as evidence, not raw, brute fact. (I'd say the only admissible evidence is what Sylvester actually wrote. If he wrote "I think of Matrix as a womb from which the determinant issues" then I'd say you could report that as brute fact).

What I suggest is that you put into a footnote exactly what you know to be the facts:

ref> Per the OED the first usage of the word "matrix" with respect to mathematics appears in (1850) J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 369. "We ‥commence‥ with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order. [If you have it, and only if you have it, can you could add e.g. Author_here year_here title here:page here states that Sylvester meant this to mean, that the determinant issues from a "womb", the Latin word for matrix. ref>

This is not OR, no more OR than quoting any other source. But without support the fantastical synthesis of "matrix" to something "issuing from a womb" certainly isn't in the quote above, at least as you've presented it. Maybe it is there in the original, but if so we need the whole quote. Bill Wvbailey (talk) 19:08, 11 April 2011 (UTC)

I am getting tired of this debate, and especially of occasionally rude language addressed at me. Somebody else should join. Your selective Websters are not doing a good job. Here is the first definition for "matrix" in OED:

1. The womb; the uterus of a mammal. Also (in later use esp. of an oviparous vertebrate or invertebrate animal): the ovary; the part of the female reproductive tract producing or storing eggs or embryos. Now chiefly hist.

with quotes such as:

(1765) Treat. Domest. Pigeons 15 "The ovary, or upper matrix of the hen, or female bird"

(1840) tr. G. Cuvier Animal Kingdom 40 "The foetus, immediately after conception, descends‥into the matrix."

That's because "womb" was a popular meaning for "matrix" at the time. And a number of RS linked above (see my first reply) should suffice, as not a single RS seem to be contradicting this. I don't see any other logical reason to continue with this historical OR on this pages. Mhym (talk) 19:29, 11 April 2011 (UTC)

You're right, this is a tiresome discussion: you've made no case whatever for your OR extrapolation of the usage of "matrix" to "generating from a womb" w.r.t. application to the above quote from Sylvester, where neither word "generating" nor "womb" appear, and you clearly are not open to the notions of evidence and reason. The only achievement here has been for you to actually do some decent research to validate the date (it's changed how many times?) and provide us with a decent source and a nice quote. My role here is complete. Bill Wvbailey (talk) 21:19, 11 April 2011 (UTC)

I got lost in this argument. At the risk of being insulted, how about this: [8]? Myrvin (talk) 18:20, 12 April 2011 (UTC)

That is a very clear and unambiguous argument in support of all the sources I listed above. I will try to fix the statement to address this. Mhym (talk) 18:47, 12 April 2011 (UTC)

(ec) I don't think you will be insulted for that. That's the smoking gun. Having seen a lot of nonsense that was passed from one author to another, sometimes over several centuries, I think I understand what Wvbailey was trying to do. We must make sure to be pretty conservative about claims that look as if they came up via a telephone game ("one could speculate that" -> "it could be that" -> "it seems likely that" -> "it is likely that" -> "it is well known that"), and this claim did look a bit fishy. But it appears to have been correct. This source is a great addition to the article. Hans Adler 18:49, 12 April 2011 (UTC)

Just dropping by to say I stand corrected in this issue. The evidence is quite convincing. Just one minor point to quibble: it would seem that when Sylvester used the word Matrix, it was as least as common in English as in Latin (in any case he did not seem to feel the need to point to a Latin origin), and I'm not sure having seen sufficient evidence that it is used with the meaning "womb" in Latin (the Merriam-Webster reference says "female animal used for breeding, parent plant"). And I still find it curious that in Latin one would use a female suffix -ix with the stem matr- that is not lacking in female connotation (if ever I need to coin a term for a new mathematical notion, I will certainly consider "patrix"). But bravo to Mhym for her perseverance in this matter. Marc van Leeuwen (talk) 11:22, 16 April 2011 (UTC)

The beginning doesn't explain what matrices are (and it's boring)

If this article is meant to inform a lay person, then read on. Sorry if this sounds too critical.

"In mathematics, a matrix...is a rectangular array of numbers..."

You've already put the reader to sleep.

This is all very concrete, and I'm sure it's useful, but I don't think it's conveying what matrices actually are. What kind of thing are they? Are they some kind of "construct"? Are they like other things? Challenge: explain matrices without describing what they look like or how they're used. What is their nature? —Preceding unsigned comment added by 71.37.42.176 (talk) 08:52, 26 April 2011 (UTC)

So, what kind of thing do you think matrices are ? We can describe how they are usually written down (as a rectangular array of numbers), how they behave (rules for addition, subtraction, multiplication by a number, multiplication by another matrix), and what they are used for (representing linear transformations and systems of linear equations, amongst other things). All of this is covered at a summary level in the first two or three paragraphs of the lead section. If we omit what they look like and how they are used, then we are left with a very abstract definition of matrices purely in terms of their algebraic behaviour. This is possible, but it requires a certain level of mathematical maturity to appreciate and understand, so it would reduce the accessibility of the article for a general reader. Gandalf61 (talk) 09:29, 26 April 2011 (UTC)

If I knew that, I'd have added it myself. :) Everything you listed makes it sound like its inventor just pulled it out of his butt. "Well, I wanted something to multiply these groupings of numbers together in this arbitrary fashion that I just made up for no reason. Hey! Look! I can also do linear transformations this way. Neat." Why do you need matrices to do linear transformations? If you don't, then what are matrices absolutely necessary for? I'm not suggesting the first sentence answer all of these, I'm just trying to get at how to think about introducing the idea of matrices. Perhaps a good way to think about this is: what role does matrices fill in math that nothing else does? What differentiates it? What is its purpose? Why would someone invent matrices? 71.37.42.176 (talk) 09:56, 27 April 2011 (UTC)

Suggestion: "A matrix, in mathematics, is a mapping that acts on vectors, which may change both their length and direction. It is represented by a rectangular array of numbers, and acts on vectors by means of an operation known as matrix multiplication. Matrices are fundamental in many areas, because they provide a relatively simple and broadly aplicable way to deal with complicated problems in mathematics, science, and engineering."

Rick Norwood (talk) 12:10, 27 April 2011 (UTC)

What you are defining is a linear map not a matrix! Matrices can be (are) used to represent linear maps, but are not themselves linear maps. A matrix is nothing more or less them a rectangular array of entries (most of the time numbers). In practical classroom treatments they two are so intimately related that the words "matrix" and "linear map" are (almost) used as synonyms, but that does not make them the same thing. In particular a matrix is not "a mapping that acts on vectors".TR 13:01, 27 April 2011 (UTC)

What you say is true, of course, but I'm trying to come up with something that doesn't put readers to sleep. How about, "In mathematics, a matrix is one way to represent a mapping that acts on vectors that makes complex calculations simpler." If not, please suggest something. Rick Norwood (talk) 15:28, 27 April 2011 (UTC)

I don't think I agree with the assertion that the current first line is particularly boring. It succinctly and correctly tells the reader exactly what a matrix is. (I think the IP who started this thread is partly victim of the confusion of thinking that a matrix is more that just a rectangular array of numbers.) Of course, it is not the most exciting phrase, but not particularly boring. Any delay in telling the reader what a matrix is, will only serve to confuse lay readers. As such I don't think the first line needs fixing.

There is room for improvement in the rest of the opening paragraph and lead though. I'm not sure about having the example of a matrix in the second line. It breaks the flow of the first paragraph making it harder to read. Would it be possible to somehow move it to an infobox? Similarly, I don't think the first paragraph of the lede should discuss notation.TR 16:00, 27 April 2011 (UTC)

I agree with TR. The very first sentence must simply say what a matrix is, with the only purpose to provide a method to tell a matrix from everything else. And the article does it. The only problem is that a matrix (in mathematics) is not exactly an array of numbers, but an array of "entries" or "mathematical objects" which are typically, but not always, numbers. Paolo.dL (talk) 17:04, 27 April 2011 (UTC)

You're right and I changed the sentence to reflect that entries are not always numbers. Saros136 (talk) 17:41, 27 April 2011 (UTC)

Thank you. I also appreciate the adjustments by Marc van Leeuwen. Paolo.dL (talk) 13:48, 28 April 2011 (UTC)

We've been so focussed on the lead, that we missed that the very first sentence of the body of the article now plainly contradicts the lead by saying that the entries of a matrix must be numbers. Of course one can find any number of textbook authors that give simplified statements like this for the sake of pedagogy. But it's not good. What to do about it? (Replacing the definition and the reference by Bourbaki(s) does not seem wise either ;-). Marc van Leeuwen (talk) 14:07, 29 April 2011 (UTC)

Well, further down, the article does have a section "Matrices with more general entries". Getting into too much detail about this at the start of the article seems like a bad idea from the perspective of accessibility. And we better write this article in such a way that an average middle schooler can understand at least the first few sections of it. May be some sort of note on the first definition that points to the section with generalizations?TR 15:01, 29 April 2011 (UTC)

I've been cleaning up the beginning to make it readable. I moved the plural parenthetical explanation to the next paragraph. I think it's definitely an improvement, but Jakob.scholbach wants to revert. Bhny (talk) 02:35, 3 May 2011 (UTC)

Sorry, I don't agree with you. I prefer the previous version, now restored by TR. It is more complete and has a more standard format and structure. Paolo.dL (talk) 13:47, 3 May 2011 (UTC)

Array = tensor? Column matrix = vector?

I am not sure that the term Array is a synonym of Tensor. Also, row and column matrices need not to be the components of linear-algebraic vectors Paolo.dL (talk) 16:18, 28 April 2011 (UTC).

It is important to keep in mind that the article's lead should match the article. For example, array data types are nowhere mentioned in the article (rightfully, I think), hence the lead should not do so either. If you disagree with this non-appearance, I suggest first working on the article, then on the lead section. Jakob.scholbach (talk) 18:50, 28 April 2011 (UTC)

Thank you for pointing this out. I agree with you. I don't really care to explain what arrays are in computer science. I only want to be sure that we don't say something wrong, i.e. that multi-dimensional arrays are (always) called tensors. Paolo.dL (talk) 01:00, 29 April 2011 (UTC)

Array and Tensor are not synonyms. The relation between an array and tensor is akin to that between a matrix and a linear map.TR 08:28, 29 April 2011 (UTC)

Arrays with more than two dimensions

I agree with Marc van Leeuwen that simlifying the intro is desirable, but possibly one of the paragraphs was over-simplifyed. In this paragraph, the expression "Arrays of values of more than two dimensions" is unclear:

Matrices with only one entry are also called scalars. Matrices with only one row or column are also called row or column vectors, as they are typically used to define the components of vectors. Arrays of values of more than two dimensions are not called matrices, but they can be interpreted as tensors.

The reader is not supposed to know the concept of "dimension of an array". I propose to modify the paragraph as follows:

Matrices with only one entry are also called scalars. Matrices with only one row or column are also called row or column vectors, as they are typically used to define the components of vectors. Scalars, row or column vectors, and matrices with rectangular shape are considered to be, respectively, zero-, one-, and two-dimensional arrays of values. Arrays with more than two dimensions (e.g. three-dimensional arrays, shaped as rectangular parallelepipeds) are not matrices and are sometimes called tensors.

I do not want to impose my (longer) version. I highly value the opinion of other editors, especially if they are so talented as those who wrote this article. I just want to point out that some readers might not understand the current version. Paolo.dL (talk) 16:45, 28 April 2011 (UTC)

Paolo.dL (talk) 16:45, 28 April 2011 (UTC)

I think the last line about arrays with more than two-dimensions can be safely dropped without detriment to this article.TR 08:30, 29 April 2011 (UTC)

I agree that is better not to mention "arrays with more than 2 dimensions", than mentioning them without explaning what a dimension is, but there's also another option: explaining it (e.g., as I proposed above). Paolo.dL (talk) 12:41, 29 April 2011 (UTC)

The sentence about "Higer-dimensional arrays" has been reinserted (at the end of the introduction). We already discussed about this, and decided to remove it. There are three possibilities:

1. we don't use, in the introduction, the expression "higher-dimensional" (or "with more than two dimensions").
2. we use it, but first we explain it (e.g., as I suggested above)
3. We use it without explaining it

I believe that the third option should be avoided. In general, too technical terms or expressions should be either avoided or explained, in an introduction. About this, I invite everybody to share their opinion. Paolo.dL (talk) 14:36, 30 April 2011 (UTC)

A scalar is a matrix?

On second thought, I am not sure that a scalar is a matrix. Of course, matrices may have only one entry, theoretically. So, a single value s (italics font) can be viewed as a 1x1 matrix s (bold font). The problem is that, if you call it a scalar, then I guess you cannot call it also a matrix. The two concepts s and s seem to be incompatible.

In the MATLAB environment (as you probably know, MATLAB means MATrix LABoratory, not MATh LABoratory), scalars are defined as 1x1 matrices, but I think this is not 100% correct in mathematics, because in mathematics the definition of scalar multiplication (scalar by vector, scalar by matrix, or scalar by N-D array) is not (always) compatible with the definition of matrix multiplication. (Indeed MATLAB uses a less strict definition of the matrix multiplication, represented by a function called MTIMES, which performs scalar multiplications when one of its operands is a scalar, but I believe this is not 100% correct in mathematics)

For instance, the product of a scalar s by a 3x3 matrix M, as far as I know, does not meet the (strict) definition of a matrix multiplication, although it can be represented by the matrix multiplication SM, where S = sI.

Also, if v is a 3x1 column vector, then sv (matrix multiplication) is impossible, while

vs (matrix mult.) = vs (scalar mult.)

Paolo.dL (talk) 11:09, 29 April 2011 (UTC)

Scalar can mean a number of things. One of them is a rank 0 tensor, which can be represented as a 1x1 matrix. In this case, one might even say that the scalar is the 1x1 matrix, since this representation does not depend on any choice of basis. Scalar multiplication (of matrix), can be thought of as the tensor product (or rather the Kronecker product) of a matrix by a 1x1 matrix. (On the last subject, why is this operation not mentioned in this article?)TR 11:18, 29 April 2011 (UTC)

Interesting, but then again, a Kronecker product is not a matrix product. I think it is safer not to mention scalars or 1x1 matrices in the introduction. Paolo.dL (talk) 11:53, 29 April 2011 (UTC)

(Hm... the text above this reply has just changed (edit conflict); curious.) Even if there might be some form of formal justification, calling a 1x1 matrix a scalar would be quite confusing in the context of this article. In the context of matrix algebra, one distinguishes scalar multiplication and matrix multiplication as different operations, and the scalar argument in scalar multiplication is a scalar, not a matrix (seems obvious). If one does want to view scalar multiplication as a form of matrix multiplication, then one needs to represent the scalar by the corresponding multiple of the identity matrix, not by a 1x1 matrix. Of course sufficiently knowledgeable people will know that there is some way (the Kronecker product) to view scalar multiplication as multiplication by a 1x1 matrix, but really, these people do not need to read this article to find that out. IMHO the lead should be kept at a quite basic/broad audience level. Marc van Leeuwen (talk) 11:55, 29 April 2011 (UTC)

Agreed, thank you. Paolo.dL (talk) 12:01, 29 April 2011 (UTC)

I agree as well, to avoid confusion we should try to use the word scalar only in one sense of the term. I do think that the article should mention the Kronecker product somewhere (not in the lead).TR 12:10, 29 April 2011 (UTC)

A scalar is an element in a field. A 1x1 matrix is not. Rick Norwood (talk) 12:09, 29 April 2011 (UTC)

Actually, a 1x1 matrix (with values in a field) is itself an element of a field. ;)TR 12:12, 29 April 2011 (UTC)

I think that the scalars can be naturally identified with the 1×1 matrices. One could say something along the lines that though scalars are not technically 1×1 matrices, the two concepts are often conflated since confusion is unlikely to arise (or something of the sort). RobHar (talk) 14:17, 29 April 2011 (UTC)

What about the previous subsection? We didn't reach a consensus yet. I invite everybody to share their opinion. Paolo.dL (talk) 12:41, 29 April 2011 (UTC)

Yes, if we have a field of scalars, then we can use that to define a field of one by one matrices with entries from that field of scalars. But if you try to avoid mentioning scalars as distinct from matrices, and define scalars to be one by one matrices, how to you finish the sentence, "One by one matrices with entries from ...". You certainly don't want to say, "One by one matrices with entries that are one by one matrices." Since you can't avoid mentioning scalars, why not do what all the Linear Algebra books I've taught out of do, and make a distinction between scalars, which are the entries in the matrices, and the matrices themselves. Rick Norwood (talk) 15:29, 29 April 2011 (UTC)

This is great

I love the original post in this thread. The anon is voicing the very natural desire to understand, by asking "Yes, but what is a matrix"! Can I ask why nobody responded with: "You cannot be told what a matrix is.. you have to see it for yourself." But seriously, the anon has a point.. after all, a matrix in the sense of this article is more than just an array of numbers or symbols.. in the same way that a dictionary is more than just a book full of words. And the opening sentence doesn't indicate that in mathematics the term "matrix" has more meaning than simply being an array of things.. I think this is the problem the anon was having. Notice that both the German and French Wikipedias mention this extra structure in their opening sentences. 137.82.175.12 (talk) 23:36, 25 May 2011 (UTC)

Element or entry redux

A SciFindScholar search for papers containing "matrix element" and "matrix entry" found 7913 and 8 papers, respectively. This is consistent with "entry" seeming bizarre in a natural science context. A Web of Science search reported 27,000 articles containing "matrix" & element", and 1800 containing "matrix" & "entry". Here, the search is confused by the many topics that place these words close together (e.g. pieces of solid in which other solids are embedded). Searches on the word pairs "matrix element" and "matrix entry" found 36 and 3 papers respectively. Another search found several papers containing "matrix entry" in the journal "Linear algebra and its applications". A direct search of this journal, that has been publishing since the 1960s, reported 4000 occurrences of "matrix entry" and 5000 of "matrix element". I have tried to accommodate these findings in a small edit. I hope it does not tread on toes. Michael P. Barnett (talk) 20:09, 30 April 2011 (UTC)

Quite generally, it is a bad idea to count hits somewhere online. Instead, one must consult authoritative sources, such as, say Serge Lang "Linear Algebra". He calls the things entries or components (p. 23). Your unilateral rename everywhere is inappropriate, I believe. Jakob.scholbach (talk) 17:47, 1 May 2011 (UTC)

What is the perspective supposed to be?

I was led to this article today by accident, and was startled by the presence of the term "matrix entry", then saw this had been questioned, and thought the questioner entitled to informed follow up. Hence the section preceding this. When I started to accommodate the preferred usage of "element", I made the minimum patching to avoid major dissonance with my experience (applications in published research, undergraduate and graduate course) of working with matrices for over 60 years -- a major newspaper article in the Guardian Weekly last week indicated a concern about the dearth of academics who contribute in their field of expertize -- but I have seen that other professionals avoid stating their credentials so I say no more on this. However, the concensus in the ratings seems to be that this is a good article, so I may be quite wrong about what it is intended to do. If so, please revert what I have done. If anyone wants a clarification of changes I made, I will be responsive. Thanks. Michael P. Barnett (talk) 21:00, 30 April 2011 (UTC)

Thanks for the very welcome information. I think scientists are more apt to use "matrix" in the geological sense, and not be bothered with "element" in the set theory sense. Mathematicians, the other way around. Rick Norwood (talk) 21:07, 30 April 2011 (UTC)

Many thanks for your comment. It encourages me to go on for a bit, albeit restricted to books I have at hand -- cannot get to library for several days. As regards "element" or "entry", I have now found in my copy of the MIT Press 1993 edition of Ito's 2-volume Encylopedia of Mathematics, section 269 (III.2) "By a matrix with elements in K ..." (where K is a ring or a field), BUT, a little later: "The element a_ik is called the (i,k)-element (entry or component).

So I think my use of element safe. If the preferred term is a big deal someone might like to check Sylvester's paper considered by Morris Klein to be the first use of the term -- in case the article does not give it, the reference is Phil Mag (3) 37 1850 363-70 (Coll math papers 1, 145-151). I have looked in about 20 other books on applied math, theoretical physics, operators, elementary abstract algebra and suchlike -- typical of someone trained in natural sciences 60+ years ago who has used special functions and linear algebra since then. They use element exclusively. Will post this and continue with some concerns, unindented. Michael P. Barnett (talk) 02:30, 1 May 2011 (UTC)

I'm not sure about this.. to a mathematician, a matrix itself is often considered as an "element". If you're referring to the individual entries in a matrix, then the word "entry" seems way less ambiguous.. because "a matrix element" could either refer to the matrix as a whole, or some specific entry in the matrix. It's probably better to use unambiguous terminology. 137.82.175.12 (talk) 23:11, 25 May 2011 (UTC)

topics commonly mentioned

Here are terms that are mentioned in the accounts of matrixes in books that are classified as mathematics (albeit elementary), physics, chemistry and engineering: relationship to determinants (the WK article on these is bizarre), addition, subtraction, scalar multiplication, matrix multiplication, distribution, commutation (and non-commutation), association, zero matrix, identity matrix, trace, transposed, singular, reciprocal, associate, symmetric, orthogonal, real, Hermitian, unitary matrices, vector space, linear transformations, equivalent matrices, bilinear and quadratic forms, characteristic equation, reduction to diagonal form, eigenvalues, eigenvectors, congruent transformations, orthogonal transformations (Margenau and Murphy, Mathematics of Physics and Chemistry).

From the Ito Encyclopedia: Kronecker products (need term direct product, too) Hamilton Cayley theorem.

From many books, matrix representation of groups.

From books on numerical analysis -- inversion and diagonalization -- major approaches, impact of high precision arithmetic. Strassen algorithms.

Field of application -- present mention of geometrical optics very selective choice. Quantum theory host of applications. Likewise statistics.

Only trying to point out that coverage is very thin and non-representative. Needs informed thought and planning to build on these comments. Many people would consider this higher priority than working in high order arrays. Hope this triggers more input from people who have taught and used matrices -- I am sure I have omitted several topics and terms as important as those I have put in. Michael P. Barnett (talk) 03:19, 1 May 2011 (UTC)

I don't know exactly what this post is supposed to mean. Most of the topics you point out are covered in the article. About the choice of applications: there are tons of applications and in the GA process we chose to present a little bit in depth a few select applications instead of just mentioning whatever is out there.

Another simple rule should be obeyed in touching up the lead section: the lead is summarizing the article. If you feel that certain topics are under-represented, rather write about them in the article (first). Jakob.scholbach (talk) 17:51, 1 May 2011 (UTC)

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

The proposal was Withdrawn.

---

Merger proposal

I propose to merge Single-entry matrix into Matrix (mathematics). Single-entry matrix as a concept is not important enough to merit an article, and could easily be mentioned within Matrix (mathematics).

This is a merger proposal was suggested by Michael P. Barnett in April, but not followed through. It makes sense so I am redoing it.

--Muhandes (talk) 08:58, 19 May 2011 (UTC)

• Agree. Rick Norwood (talk) 12:14, 19 May 2011 (UTC)

• I don't find the content to be worth merging (or even really keeping). "Single-entry matrix" doesn't seem to be a term in wide use, and it's clearly not important enough to be treated in the main matrix article. Sławomir Biały (talk) 12:15, 19 May 2011 (UTC)

I'm not sure what you are saying. If it is important it should have an article. If it is not, it should not have an article. Are you saying that it is not important, so it should not have an article? --Muhandes (talk) 12:21, 19 May 2011 (UTC)

It doesn't seem notable enough for its own article, and it's not an important enough topic to be discussed in a main article about the subject. The main article should be even more selective than our general criteria, since there are potentially thousands of topics that could be mentioned (but quite rightly are not). Sławomir Biały (talk) 15:10, 19 May 2011 (UTC)

• From a purely practical standpoint, it is easier to merge articles that to delete one. I think the article single-entry matrix can be merged into a single sentence in this article. Rick Norwood (talk) 13:05, 19 May 2011 (UTC)

The entire Single-entry matrix article is one sentence, so of course, it would be merged as (at most) one sentence. --Muhandes (talk) 13:18, 19 May 2011 (UTC)

I disagree with merging that here. The article on matrices needs to focus on what is most important. Matrices with one entry are hardly ever important, which is why the article cannot tell much about them. If we start adding every possible definition related to matrices in this article we end up with something like list of matrices. If anything, single-entry matrix might be merged (or redirected) to list of matrices. Jakob.scholbach (talk) 13:53, 19 May 2011 (UTC)

The literature uses the expression "single-entry matrix" to indicate a matrix with a single non-zero entry. I know that we need to respect the literature, but this terminology is none the less highly misleading: obviously, the "single-entry" matrix does not have a single entry (see example)! I disagree with merging, but I suggest to rewrite the definition: "A matrix with a single non-zero entry is sometimes improperly called a single-entry matrix, and denoted as follows:...". However, we might need a section about "special matrices", (with reference to main article: List of matrices) in which we shortly define some of the most important and most frequently used kinds of matrices, for instance:

• square matrices
• diagonal matrices
• identity matrices
• sparse matrices
• block matrices
• column and row matrices

The "single-entry" matrices are, in my opinion, not important enough to be listed in this section. They should, however, to be listed in List of matrices. And we don't need to list all these matrices in the introduction. Paolo.dL (talk) 14:09, 19 May 2011 (UTC)

disagree with merger and agree with Paolo.dL. "single-entry" should just be in List of matrices Bhny (talk) 17:07, 19 May 2011 (UTC)

• Withdrawn. Some good arguments were made, and I see consensus will not be possible anyway, so rather than continuing the argument I'm going to withdraw my proposal. If someone else feels continuing this discussion might go anywhere feel free to open the discussion. --Muhandes (talk) 06:09, 20 May 2011 (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.

another idea

I know the above discussion was closed, and maybe this isn't the appropriate place to bring it up, but Single-entry matrix should probably just be copy and pasted into Sparse matrix with a redirect.. and maybe with a better definition (i.e. not insist the single entry is a 1). Just sayin'.. 137.82.175.12 (talk) 23:05, 25 May 2011 (UTC)

Reply to "This is Great" section above.

I agree with the comment in the "This is Great" subsection. While there are examples of matrices that are simply arrays, the Alexander matrix of a knot for example, in most cases matrices are linear functions, and the lede should say that. The question is, how to say it in a way that is clear to the lay reader. Rick Norwood (talk) 12:17, 26 May 2011 (UTC)

In most cases matrices are used to represent linear functions. In fact, this use is so common that people often misuse the terminology and not make the distinction between a linear map and the matrix that represents it in a certain basis. This is a form of jargon, which should be avoided as standard use in an encyclopedia, although this jargon should be mentioned.TR 12:30, 26 May 2011 (UTC)

I agree completely, but how do we explain that to someone who doesn't know any mathematics? Rick Norwood (talk) 12:34, 26 May 2011 (UTC)

I've done some work on the lede, fixed a few typos. The strongest impression I got, while editing the lede, was that it gets too technical too quickly. I think a lot of it can be cut. Advanced topics are better covered in the body of the article, or in articles of their own. But I did not want to do too much at one time. There is still a lot of work to be done. Rick Norwood (talk) 13:04, 26 May 2011 (UTC)

A recent addition to the lead leaves me speechless: "One major use of matrices is in linear algebra, where they play a role that is in some ways similar to the role played by functions in algebra." My first impulse was to throw out the sentence as original research, but it might be more instructive to ask first what on earth this statement was intended to convey. What particular role do functions play in algebra? Unless they are some kind of morphisms, algebra deals very little with functions, much less then analysis does in any case. And how does this role compare with that of matrices? The main virtue of matrices it that they can be written down, which functions cannot in general. Of course matrices, like many other objects in mathematics, can be defined as functions (from a rectangular index set to some set where the entries live), but this does not seem to be what the sentence is pointing to. There are some other similar interpretations possible of this sentence, but none that is in any way convincing. Please clarify. Marc van Leeuwen (talk) 06:47, 28 May 2011 (UTC)

I'm at a loss to understand what you are getting at. "algebra deals very little with functions"??? I was, of course, using algebra in the sense a layperson would understand the word: the subject taught in high school in which most people are first introduced to functions, but even in abstract algebra functions certainly play a major role. The article as it stood gave no indication what matrices were used for. They are most often used to represent linear functions acting on vectors, analogous to y = 2x, a linear function acting on real numbers. Remember, we are writing for a reader who has little or no mathematics beyond what is taught in secondary school. Mathematicians already know what a matrix is. The fact that most functions cannot be written down is a technical abstraction far beyond the subject matter of this article. If you really want to go there, most matrices cannot be written down, because most real numbers cannot be written down. When I get to your sentence that begins "Of course matrices..." I cannot make heads or tails of it. You seem to confuse a vector space (the codomain of a linear transformation) with the field of scalars (some set where the entires live).

In any case, I hope the explanation that "algebra" means the subject taught in high school, not abstract algebra, explains the purpose of the sentence you object to. Rick Norwood (talk) 13:59, 28 May 2011 (UTC)

I undid the recent revert. Please see the discussion above. In a first course in Linear Algebra, matrices are introduced as a way to represent linear functions on a vector space. The only problem with saying just that is, of course, that they only represent a linear function with respect to some choice of a basis, for both the domain and the codomain, just as an array of numbers only represents a vector with respect to some choice of a basis. But we cannot get into all this in the lede -- already we hear objections that a) the lede is hard for non-mathematicians to read and b) the lede does not say what matrices are good for.

We can say what a matrix is. It is a rectangular array of symbols, usually numbers. But I have not been able to come up with a way to say precisely why the study of matrices is so important, something that I didn't understand as an undergraduate (to my sorrow) and something many of my students need to have explained. Certainly, the idea expressed above by Marc van Leeuwen, that they "can be written down", is close to the mark, but conveys nothing to the layperson, who would naturally ask, "Who cares? Lots of stuff 'can be written down'." The best I've been able to come up with is to make an analogy with functions, something most readers have at least heard of and may understand the importance of. Rick Norwood (talk) 14:16, 28 May 2011 (UTC)

OK, thanks for the reply which clarifies a lot (and which I'm reading somewhat late). The confusion was not so much about the word "algebra", in so far as high school algebra is actually a part of algebra, but about the word "function". I was thinking of such functions as exponential, trigonometric, polynomial functions and their compositions, which (1) as such do not play a very central role in algebra, and (2) the role they do play is in no way similar to that of matrices. Saying y = 2x is a linear function acting on real numbers shows what you meant, but is a very sloppy use of language. I'd say y = 2x is a linear equation involving two (real valued) variables x,y. The function that is involved here is the one from real numbers to real numbers that (assuming the usual convention for variable names) doubles its argument. Now saying that matrices play the same role as functions is still unclear to me. Matrices are used to represent linear transformations, so functions supposedly similarly represent something, but what? The only thing I can think of here is some real-world relation between values that is modeled by the function; however their relation is rather different than linear transformation–matrix. My guess is that you meant to say that matrices play a role relative to linear transformations in linear algebra that is similar to that of expressions relative to functions in (high school) algebra. You may ignore the "Of course" sentence above, which was just a wrong guess about the suggested relation between matrices and functions, but in case you wonder, the explanation is this: a m × n matrix A = (a_i,j) with rational entries can be defined as a map (function) {1,...,m} × {1,...,n} → ℚ that sends (i,j) ↦ a_i,j, in other words the matrix itself (as is proper for a definition) rather than the linear transformation it represents. Marc van Leeuwen (talk) 09:34, 29 May 2011 (UTC)

The last sentence of the first paragraph is currently extremely vague. I fear it may be mystifying to lay and expert readers alike. What wrong with just saying:

"A major application of matrices is to represent linear transformations of vectors in linear algebra."

This has the advantages of saying exactly what matrices are used for. Of course, we may have the issue that a lay reader has no idea what a vector or a linear transformation is. But even that reader will get more information from this sentence, then the current enigma. (Any person with normal reading comprehension should be able to distill the following from this sentence: Matrices are used in a field called "linearv algebra" to represent some kind of relation called a "linear transformation" between objects called vectors. To help the lay reader it may be an idea to give an explicit example of how a matrix can represent a 90 degree rotation of the plane. Although we should also avoid the pitfall of trying to explain everything in the lede.TR 15:31, 28 May 2011 (UTC)

Consider this statement: "A major applications of matrices is to represent wibblefubwilda of tribbles in jabberwocky." That's what the mathematically correct formulation is going to sound like to a lay reader. True, there are hyperlinks, but a frustrated reader is unlikely to follow them. I understand that we can't explain everything in the lede, but we need to give some idea of what matrices are good for. The sentence "matrices represent functions" is mathematically correct, and focuses on the first application of matrices most students see. We could say exactly that, but it seems to me better to say something that tells the lay reader that matrices are similar to the functions they learned about in high school. How about this: "Matrices are used to represent functions, similar to the functions taught in secondary school, functions which are important in many fields, functions which are most easily expressed in matrix form." Rick Norwood (talk) 15:45, 28 May 2011 (UTC)

Lay readers are not helped at all by being vague. Moreover, referring to what students learn in XX school is not helpful to the general audience, since there is a large part that both has no clue what matrices are and has no clue what is taught when in what school.TR 17:08, 28 May 2011 (UTC)

Couldn't agree more with TR. I reworded that bit and blended it with the remainder of the lead. Jakob.scholbach (talk) 17:18, 28 May 2011 (UTC)

I can understand Rick Norwood's objection to the sentence "A major application of matrices is to represent linear transformations of vectors in linear algebra", because not everybody will understand it. But this is a serious encyclopedia, not a friendly textbook written for high-school students. I think this sentence gets to the core of what matrices are, in a concise and (compared to most mathematics articles) clear way, and so it should be one of the first sentences of the article.

Currently I read the first few sentences as: "Matrices are just rectangles filled with things; look here's one (example).. see they're not too scary!" Frankly, the opening paragraph is missing the point. I think TR's suggested sentence is far superior. We should not be dumbing the first paragraph down to the point where the fundamental purpose of matrices is lost. 137.82.175.12 (talk) 22:06, 30 May 2011 (UTC)

Suggestion for first paragraph

I do think it is a good idea to have the prime use of matrices, representing linear transformations, as early as possible. How about the following suggestion for the first paragraph?

${\begin{bmatrix}1&9&13\\20&55&6\end{bmatrix}}$

An example of a matrix with six elements.

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its components, elements, or entries. A major application of matrices is to represent linear transformations of vectors such as rotations. For example a 90 degree rotation of the plane transforms the vector (1,0) to (0,1), and (0,1) to (-1,0), this can be represented by the matrix,

${\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.$

The first row expresses what (1,0) is transformed to, and the second row expresses where (0,1) is send to.

The current picture would be moved to the notation section. The use of row vectors rather than the more common column vectors is not ideal, but does make it easier to mention the vectors inline.TR 17:41, 28 May 2011 (UTC)

I don't mind moving the linear transformation bit up, but it needs to be in line with the rest of the lead. Moving just one sentence as someone did earlier simply creates a disconnected text. Also, sums of matrices are easier to understand than products (and are also treated earlier in the article). I don't have a strong opinion on the picture. I disagree, though, with putting these details about the rotation matrix. First of all, this is practically impossible to understand for a newbie, and also is unnecessarily detailed. I could imagine having a picture that shows the linear transformation (but I would use a matrix all of whose entries have different absolute value) attached to a 2x2-matrix could help in the lead. Jakob.scholbach (talk) 20:42, 28 May 2011 (UTC)

I find the current version acceptable, and agree that the version in a box above is too complicated. There is still a small technical problem: y = 2x + 3 is a linear function that does not generalize to a linear transformation, but I can live with that. It would be ideal to have a brief animation that illustrated the action of a matrix on a geometric object such as a cube, but that is beyond my ability to program? Anybody? Rick Norwood (talk) 00:50, 29 May 2011 (UTC)

From Wikipedia Manual of Style (layout)

"The lead section should briefly summarize the most important points covered in an article in such a way that it can stand on its own as a concise version of the article. The reason for the topic being noteworthy should be established early on in the lead. It is even more important here than for the rest of the article that the text be accessible. Consideration should be given to creating interest in reading the whole article. (See news style and summary style.) This allows editors to avoid lengthy paragraphs and over-specific descriptions, because the reader will know that greater detail is saved for the body of the article." Rick Norwood (talk) 12:31, 31 May 2011 (UTC)

I think most editors are aware of that, so I'm a little confused as to what point you are try to make by quoting that here.TR 12:53, 31 May 2011 (UTC)

Sorry. I was responding to this, above:

"I can understand Rick Norwood's objection to the sentence "A major application of matrices is to represent linear transformations of vectors in linear algebra", because not everybody will understand it. But this is a serious encyclopedia, not a friendly textbook written for high-school students. I think this sentence gets to the core of what matrices are, in a concise and (compared to most mathematics articles) clear way, and so it should be one of the first sentences of the article. 137.82.175.12 (talk) 22:06, 30 May 2011 (UTC)"

Anyone who comes to this article trying to learn what a matrix is used for will not understand "linear transformation of vectors in linear algebra". This explanation fails to meet the requirement of accessability. That Wikipedia is "not a friendly textbook written for high-school students" does not justify confusing almost all readers from the outset. It may be that 137 underestimates just how forbidding most people find mathematics. I was recently asked, by an accountant!, what mathematics is good for. I started on some high-falutin' explanation when she interupted, "I want you to know I don't think all those little x's and y's really mean anything." This is easy to make fun of, but the person is intelligent, highly educated, and a well-paid professional. Maybe no Wikipedia article on mathematics will reach the average reader, but it would be nice to reach at least some readers. Rick Norwood (talk) 14:01, 31 May 2011 (UTC)

I also consider TR's suggested "A major application of matrices is to represent linear transformations of vectors in linear algebra." to much in-your-face, but the current "A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x." looks fairly good to me. Jakob.scholbach (talk) 19:40, 31 May 2011 (UTC)

P.S. To be honest, I don't think these elongated discussions about the lead of this article help the article improve much. It is certainly not a perfect article or lead section, but at least a GA. There is much worse than this we should focus on, I think. Jakob.scholbach (talk) 19:40, 31 May 2011 (UTC)

Although the opening paragraph should address a broad audience, I don't think this is achieved by trying to explain terms like "linear transformation" that are (one hopes better) described in their own article. If the essence of what matrices are useful for can be explained without mentioning linear transformations, then this should be done, but if knowing about linear transformations is a prerequisite for understanding the use of matrices, then the lead should link to the notion and assume the reader understands it. I cringe at sentences that mention for instance "entries in a field" and then haste to add "like the real numbers" as if that really helps the readers put off by the term; "generalizations of linear functions such as f(x) = 4x" suffers from the same problems (one who is unfamiliar with linear transformations is unlikely to understand what kind of generalizations these could be, and probably just feels even less comfortable). To add something constructive, I think one could start (after the initial definition) by saying something like that matrices serve to treat the set of entries they contain as a unit, when these entries are used together to express linear relations between sets of values. A first example could be the collection of coefficients in (the left hand side of) a system of linear equations (after it has been brought into a standard form). This is much less abstract than talk about linear transformations; people can immediately spot why a rectangular set of coefficients is involved. One could then add that "more generally matrices provide a concrete representation of linear transformations" or something similar. Then one could add that operations can be defined on matrices depending on their interpretation (for instance in systems of equations it is natural to consider interchange of equations, or addition of one equation to another), and that notably composition of linear transformation gives rise to matrix multiplication. But in fact I'm more thinking about an introductory section than about the lead here; the lead should really be in a summary style and not involve detailed examples. Formulating a good lead is a difficult balancing act, and I fully agree that we are putting too much effort into doing that rather than improving the article first. Marc van Leeuwen (talk) 04:12, 1 June 2011 (UTC)

I already said above that the sentence "...linear transformations, that is..." looks good to me. I agree that is is time to move on. Rick Norwood (talk) 12:00, 1 June 2011 (UTC)

Wikimedia matrix reflects a LaTeX bug

${\begin{bmatrix}1&9&13\\20&55&6\end{bmatrix}}.$

${\begin{bmatrix}1&2&3&4&5&6&7&8\\a&b&c&d&e&f&g&h\end{bmatrix}}.$

${\begin{bmatrix}1&2&3&4&5&6&7&8&9\\a&b&c&d&e&f&g&h&i\end{bmatrix}}.$

${\begin{bmatrix}1&2&3&4&5&6&7&8&9&10\\a&b&c&d&e&f&g&h&i&j\end{bmatrix}}.$

${\begin{bmatrix}1&2&3&4&5&6&7&8&9&10&11\\a&b&c&d&e&f&g&h&i&j&k\end{bmatrix}}.$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \setcounter{MaxMatrixCols}{14} \begin{bmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\a & b & c & d & e & f & g & h & i & j & k & l & m & n\end{bmatrix}. \setcounter{MaxMatrixCols}{10}}

$\left[{\begin{matrix}1&2&3&4&5&6&7\\a&b&c&d&e&f&g\end{matrix}}\ \ {\begin{matrix}8&9&10&11&12&13&14\\h&i&j&k&l&m&n\end{matrix}}\right].$

MathInclined (talk) 00:08, 16 June 2011 (UTC)

— Preceding unsigned comment added by MathInclined (talk • contribs) 00:05, 16 June 2011 (UTC) 

This is no bug, it's a feature. And not only of texvc, but also of the original amsmath. Since the number of columns is not predefined, 10 columns are always allocated and the actual number used is determined during the interpretation. To use more produces an out of bounds error. There is a special macro in amsmath to increase the number of preallocated columns. But due to the wiki-structure, that probably won't work here. A way around is the blockwise construction of large matrices. Or the use of the old-fashioned array environment.

$\left[{\begin{array}{cccccccccccc}1&2&3&4&5&6&7&8&9&10&11&12\\a&b&c&d&e&f&g&h&i&j&k&l\end{array}}\right]$

--LutzL (talk) 08:44, 16 June 2011 (UTC)

Characteristic polynomial's variable?

Small miff in the article, why does the polynomial change from λ to t? Would we be better off editing "The function pA(t) = det(A−tI) is called..." into "The function pA(λ) = det(A−λI) is called..." or is there a common use I'm missing? My relationship with eigenvalues has been long-distance at best, I'm afraid. :) 91.177.218.85 (talk) 11:03, 17 January 2012 (UTC)

Hi, the polynomial does not change variables. The polynomial is defined using the (unspecific, general) variable t. λ stands for a specific value, a root of the polynomial. So p(t) is the polynomial and p(λ)=0 or det(A-λI)=0. -- That does not say that your version is not equally correct, but it would not be an improvement of the article.--LutzL (talk) 12:14, 17 January 2012 (UTC)

Functions and function spaces

This section was deleted with this edit using the one-line comment: "remove rather confused section on a subject that is already covered in infinite matrices section above". Aside from the fact that there is nothing confused or confusing about this section, it certainly is not covered by the section on infinite matrices. As indicated by the sources supplied, it relates to the connection between function spaces and matrices, and as a particular example, to the connection this provides between numerical analysis, operators upon functions, and matrices. Of course, there is a connection to infinite matrices in the case of an infinite dimensional set of basis functions, but the subject here is larger and not coextensive.

If anything, the section on infinite matrices is less transparent than this one.

This function-space section is short, and of course much more could be said. If the links to other WP articles and the sources provided are not felt to be sufficient guidance to these topics in this overview, then the section should be enlarged, not deleted. Brews ohare (talk) 02:24, 19 February 2012 (UTC)

I have to agree with this edit that it does not belong. It is just an extension to an existing section rather than some new application, so should be linked to it and not separate. But mostly it is so poorly written it is impossible to understand. It barely mentions matrices, consists half of unencyclopaedic questions and sentences of links, and even where it tries to explain it is so badly written it's difficult what it's trying to get at. This is a good article and it falls far below the standard of writing expected for such, so does not belong.--JohnBlackburne^words_deeds 02:41, 19 February 2012 (UTC)

Perhaps you could suggest some revisions, or attempt writing something yourself? The objective is to provide guidance to the other articles on WP without going into massive detail or arcane jargon, as is so common in the math article of WP, and to provide some sources, as is so uncommon in the math articles on WP. Brews ohare (talk) 04:07, 19 February 2012 (UTC)

The section was very badly written, and consisted mostly of very confused statements. The only relevant part to this article is that infinite matrices can be used to represent linear operators on infinite dimensional vector spaces (such as function spaces). This is already covered.TR 10:16, 19 February 2012 (UTC)

It would be helpful if you all could provide specifics, rather than generalities. For reference the section is reproduced below. I'll make some suggestions under it. Brews ohare (talk) 17:04, 19 February 2012 (UTC)

Functions and function spaces

Matrices have application in connection with generalized Fourier series, and their generalizations. The basic idea is that a basis set of functions {φ_n} can be used to form functions ψ = Σ c_n φ_n that are superpositions of the basis set. This relation sets up a correspondence ψ → {c_n} that can be viewed as a column vector c. Then an operation on the function ψ results in a new function, say χ, and therefore results in a new superposition χ = Σ d_n φ_n and a new correspondence χ → {d_n} to the new column vector d. The operation is then represented by a matrix, say M:

$d_{m}=\Sigma _{n}M_{m,n}c_{n}\ ,\ \mathrm {or} \ \mathbf {d} ={\boldsymbol {M}}\cdot \mathbf {c} \ .$

Needless to say, there are a number of mathematical requirements to make sure this idea works. For example, these questions are addressed:

1. What functions ψ can be expressed by the basis set {φ_n}? This topic is that of completeness of the basis functions.
2. What limitations are placed on operators upon these functions to insure they result in new functions that also are expressible using the basis set {φ_n}?
3. What meaning attaches to infinitely many basis functions (n → ∞)?
4. What are the relative merits of one basis compared to another?
5. How are the elements M_m,n corresponding to an operation determined?

These and other issues are topics in the theory of function spaces like Hilbert space, Banach space and the like.

These ideas have important application to numerical analysis where the basis functions are used to approximate differential equations with matrix equations, for example, using the Galerkin method to introduce powerful techniques like the finite element method. The basis functions in such methods often are chosen to be localized to small regions, for example, the so-called hat functions, or in one dimension, the triangular functions.

References

Discussion

Using the above, let me address your comments and request your assistance;

Comment: "The only relevant part to this article is that infinite matrices can be used to represent linear operators on infinite dimensional vector spaces (such as function spaces). This is already covered."

Although this seems to be how this section comes across, that is not an accurate portrayal of what I wished to say here. There are several points being made:

1. Operations in function space can be mapped into operations in a vector space using a basis set of functions. This point is not made elsewhere in the article. It need not be an infinite basis as you all seem to think; for example, in group theoretical treatments of atomic and molecular spectra finite sized matrices are used exclusively based upon matrices found using a limited number of orbitals.
2. That point established for the reader, I imagined that a reader needed to be cautioned that there were restrictions upon this process of converting operators on function to matrices. I did not feel it was the place to delve into these issues, so I posed them as a list of some questions the theory provides, and provided links to WP articles where answers and further discussion could be found. That seemed appropriate in this article, which is not about function spaces.
3. Finally, the important application of matrices to numerical analysis is brought up. That is most commonly framed by using a particular basis set that consists of function localized to a small region. This approach generalizes the older one based upon finite difference approximations. It is an immediate outgrowth of the earlier discussion of how a basis leads to matrix representation of operators on functions, in particular differential operations. There is nothing about numerical analysis in the article on matrices, although it is a very major use of matrices.

The above discussion is supplemented with two really excellent books that cover much more than the points they are cited in connection with.

I have set out my objectives here, which seem to support the claim that this work is not simply an unnecessary elaboration of the section on infinite matrices.

Comment: "The section was very badly written, and consisted mostly of very confused statements."

Of course, every reader will have their opinion of how well the ideas have been expressed. It would be helpful if the "confused" ideas were identified. My suspicion, and only that, is that the brief outline of how matrices are related to operations on functions was simply not couched in a sufficiently technical manner for a mathematician. I expected that reaction from such readers, and hoped the list of questions that a mathematical reader would naturally raise would assuage these concerns without bogging matters down in detail that can be found elsewhere. This approach is perhaps unusual on WP, especially in math articles that tend to be framed in jargon and without reference to background material for the uninitiated. However, I do not think there is anything confusing here. I await your detailed discussion. Brews ohare (talk) 17:04, 19 February 2012 (UTC)

Your point 1 is already very confused. Functions spaces ARE vectors spaces (something which isn't that relevant to this article) therefore operators on functions spaces are operators on vector spaces and can be represented as (infinite) matrices. That sometimes a cutoff can be introduced to approximate an infinite matrix by a finite one isn't as much an application of matrices as it is of fourier theory.

Your point 2 demonstrates that you still have a limited grasp on encyclopedic writing. Which is to bad, because it means that you are likely to continue doing more harm than good.

I have not interest in discussing this with you any further. (As the past has proven that those discussions are seldom productive.)TR 21:02, 19 February 2012 (UTC)

Timothy: It is unfortunate that you have decided further discussion will not be productive. I am unsure what the basis of that decision may be. In case some response to your remarks might be of interest to others, I'd say the following. Your sharp remark that "function spaces ARE function spaces" and using that remark as evidence that the proposed contribution is inaccurate is first of all unkind as the remark you address is not part of the proposed contribution. The proposal points out how a function can be mapped into a column vector of coefficients and makes no statement whatsoever about functions and their relation to vector spaces, which is not pertinent to the description of how correspondence of operators to matrices arise. Your later observation regarding finite matrices as cut-offs of infinite ones simply is incorrect, as finite matrices are perfectly acceptable in some applications without introducing approximation. And again, this remark is irrelevant to the discussion of the proposed contribution. And finally, your peremptory dismissal of my contributions and forecast of their ill effect now and in the future are uncalled for, uncivil, and a breach of the decorum needed for cooperation in building the encyclopedia. Brews ohare (talk) 04:26, 21 February 2012 (UTC)

Timothy, your remarks are not directed at improving the proposed contribution, and do nothing to discredit it in its present form. Brews ohare (talk) 04:31, 21 February 2012 (UTC)

Certainly it's impossible for me to help you improve what you've written: to do so I would have to understand it and I can't (and I have a degree in maths). More generally it is not the purpose of WP to help you improve your mathematics writing. If you are not experienced enough in a topic to write clearly and well on it then you should not be editing that topic; there are many other topics, and no shortage of articles in need of improvement.--JohnBlackburne^words_deeds 17:24, 19 February 2012 (UTC)

John, clearly the objective is not my improvement, but a collaborative effort among us all to make an encyclopedia. It is indeed unfortunate that you do not understand this subject, or at least not sufficiently to understand this very simple presentation, nor to write "clearly and well" on this topic. However, you could bring some perspective forward as to the confusion this article raises in your mind as a lay reader. Brews ohare (talk) 18:04, 19 February 2012 (UTC)

I understand matrices very well, and use them in my work on a regular basis, and understand the article as it now is. But I could not understand what you had written.--JohnBlackburne^words_deeds 20:20, 19 February 2012 (UTC)

To repeat, perhaps you can bring forward some specific matters that you did not understand. For example, do you find confusing the point that subject to some requirements examined later, a function can be expressed as ψ = Σ c_n φ_n?

Or perhaps you found confusing that an operation upon the function ψ could lead to another function χ?

Or, perhaps that the same kind of expansion χ = Σ d_n φ_n under appropriate circumstance could be used for χ?

Or, perhaps that a matrix could be used to map the {c_n} into the {d_n}?

Or, perhaps that when such a matrix mapping exists, it provides a representation of the operator?

Or, perhaps rather than have the mathematical requirements pointed out as questions for the reader to pursue in other WP articles, you would like more detail in this paragraph?

John, just where does there arise confusion in your mind? Brews ohare (talk) 03:56, 21 February 2012 (UTC)

Comment: Agree with the removal of any expanded details about infinite matrices in this article. That's a different subject under Hilbert spaces and suchlike. This article should confine itself in the main with finite matrices. Dmcq (talk) 12:14, 21 February 2012 (UTC)

Dmcq: Exactly where are you coming down here? The proposal refers to finite matrices, but perhaps you are not aware of that? I don't know if you support the proposal deliberately or by accident. Brews ohare (talk) 15:42, 21 February 2012 (UTC)

I don't know what you mean by 'proposal' here. The text refers to generalized Fourier series, Hilbert spaces and Banach spaces. They are all typically infinite dimensional. Then it jumps to numerical approximation with no clear link. It is messy with no clear content and seems to be mainly about infinite dimensional matrices. The content should use sources that talk about something relevant to the topic. Dragging in infinite dimensions is not relevant at the level of this topic except in a special section about that. The applications section in particular should deal with finite matrices and very clearly show how they are used. Dmcq (talk) 18:40, 21 February 2012 (UTC)

Infinite matrices certainly belong as a subsection in this article. While Brews did make a passing comment about infinitely many basis elements, that does not appear to be the main thrust of his suggestion. The main problem here is that the deleted content did not actually go beyond previously presented material. It was removed because it was superfluous. There may be a nugget or two which would go well at the end of the infinite matrix section. Rschwieb (talk) 18:48, 21 February 2012 (UTC)

Comment: Agree with removal of section "Functions and function spaces". It did not add anything new, it seems mainly to be a lot of name-dropping. It was also quite vague. Rschwieb (talk) 15:59, 21 February 2012 (UTC)

Rschwieb: You've said that my proposal offers nothing that is not already in the article. I thought the reference to numerical analysis and to matrices as related to M_ij=(φ_i, O φ_j) was something not found in the article. Perhaps you see connections I don't see with the rest of the article. Could you point them out?

You also refer to "name dropping", which I'd guess refers to the mention of completeness? I thought discussion about that would be a digression here that duplicates material better found in Hilbert space or elsewhere. Maybe you could elaborate here as well? Brews ohare (talk) 17:21, 21 February 2012 (UTC)

It's unreasonable to try to be complete within a humble subsection. That section was destined to be no longer than it already was. As commented before, the removed content said little more than the last two-sentence paragraph of the infinite matrix section. There is no reason to make an entirely new section for a finite dimensional special case. Perhaps you can fashion a few gems to add onto the end of the infinite matrix section. Mathematically trained editors might have found some phrases to be (mathematically) nonstandard (such as "superposition", which appears to mean "linear combination"). Also mentioned before: the list of five questions is unencyclopedic. There appear to be sufficient references to numerical methods without the one you added. Rschwieb (talk) 18:23, 21 February 2012 (UTC)

I misread what you meant by completeness. No, I was referring to the crowd of cryptic special terms: the Galerkin method, the finite element method, the so-called hat functions, the triangular functions, Hilbert space, Banach space, generalized Fourier series. While it's good to give examples, this magnitude of exotic characters can be distracting. Rschwieb (talk) 18:36, 21 February 2012 (UTC)

Comment: Since Matrix Theory links here, I think more theorems should be included. Byronchen150199 (talk) 05:49, 13 August 2012 (UTC)

Summary: Several editors (Timothy Rias, Blackburne, Rschweib, and Dmcq) have objected to the proposed inclusion of the subsection Functions and function spaces on the grounds that this material duplicates what already is present in the article. In an abstract sense that is true for a set of functions satisfying conditions like (f+g)(x) = f(x)+g(x) and (cf)(x)=f(cx) is a vector space, which is the focus of the article Matrix (mathematics). However, there seems to be no reason to refuse to spell out the particular application of matrices to function spaces by noticing (i) that a set of basis functions serves to map a function into a column vector, (ii) a set of basis functions serves to map an operator O on a function into a matrix M_ij=(φ_i,O φ_j).

There also is no reason to avoid pointing out that the numerical solution of differential equations can be accomplished using matrices arrived at by the use of localized basis functions.

The resistance to these proposals has not taken the form of suggesting how the article might adapt to include explicit reference to these extremely important applications, but has instead taken a defensive stance that the present proposal to mention functions explicitly is not just unacceptable but unnecessary, and consequently the changes it suggests are not needed. It appears that ghost of Hardy is present here, and that the only "good" math is pure math without application. Brews ohare (talk) 16:26, 23 February 2012 (UTC)

With respect that is mischaracterising and ignoring my objections. But whatever the reasons there are four editors now that have objected to your additions on multiple grounds which you have largely not addressed. You might take this opportunity to recognise that the consensus is against you and consider whether further effort arguing over this might be better spent elsewhere.--JohnBlackburne^words_deeds 16:36, 23 February 2012 (UTC)

John, I have made a detailed attempt above to elicit just what it is you find confusing about the proposed text, and you have not replied. I do see the consensus, although it appears to me to be driven by a certain aesthetic and not by reaction to the missing content of the article Matrix (mathematics). Brews ohare (talk) 16:43, 23 February 2012 (UTC)

Well my resistance has been because the addition seemed too far from the main thrust of the article. Things like Galerkin methods might be worth sticking into a see also or stuck into the note about finite element methods but this article should not have loads of stuff about things which are not related to the topic. I think this has been confusing 'uses matrices' and 'is about matrices' Galerkin methods are not about matrices. Any mention would have to be in the application section and we should not go into much detail about the innards. As the WP:OR guideline ssays "To demonstrate that you are not adding OR, you must be able to cite reliable, published sources that are directly related to the topic of the article, and directly support the material being presented (and as presented)." anything much beyond that goes into WP:COATRACK territory. If I was pedantic I would ask for a citation saying something like 'Galerkin methods are an application of matrices' but I'm not asking for that. I just want the article to stay reasonably on topic. Dmcq (talk) 16:40, 23 February 2012 (UTC)

@Brews: Nobody has said we are opposed to every single thing you wrote, we have just said that it would be better introduced in other sections. You keep asking "what is wrong/confusing with what I wrote" and I don't see any real acknowledgement on your part of our answers to your question. The answers were (again): the suggested passage borders on straying too far from the title topic; some quirky phrases (which are probably a product of your background, but this is a math article after all); the list of questions is unencyclopedic and can't be included; the added material is only superficially "new" and doesn't deserve its own section. You acknowledged the last point, but mainly with a rant about the oppression inherent in the system. What about the other points?

To make my earlier suggestion more plain, I think it would be a good idea to take the tangle of applications you mentioned in the suggested paragraph, locate places where they fit in, and use the example to spruce up that place. We could tackle these new additions one at a time. Rschwieb (talk) 17:49, 23 February 2012 (UTC)

I have no further interest in this article. I have made an effort to point out its shortcomings and written a subsection to meet them. No attempt has been made to suggest a means to address these problems, and only argument is forthcoming. It's a wasted effort. Brews ohare (talk) 01:32, 24 February 2012 (UTC)

No attempt except this (twice), which I will now underline so it cannot go overlooked: I think it would be a good idea to take the tangle of applications you mentioned in the suggested paragraph, locate places where they fit in, and use the example to spruce up that place. We could tackle these new additions one at a time. Every conceivable step has been taken to accomodate you cooperatively. The offer still stands if you decide to come back. Rschwieb (talk) 21:01, 27 February 2012 (UTC)

Definition

In the introduction an explanation in terms of rectangular arrangement is already given. This however is not a definition, but merely a description to make the notion understandable. This also holds for the so called definition in the article. The definition I gave, however, is a formal definition, which in my opinion should not fail. Nijdam (talk) 21:22, 4 June 2012 (UTC)

Frankly I don't know where to start with your definition. First this is not an advanced mathematics article but a very elementary one, so definitions should not depend on an advanced understanding of mathematics or use of symbols. Second your definition was wrong: matrices are not restricted to just real and complex numbers but can contain any sort of thing. E.g. one definition of the cross product is the determinant of the matrix with i, j and k in the first row, real numbers in the other two rows. Third, ignoring the difficulty and correctness of it, your version had some basic grammatical errors that made it incomprehensible.--JohnBlackburne^words_deeds 22:21, 4 June 2012 (UTC)

The objections JohnBlackburne raises are essentially those I had in mind when I reverted. Rschwieb (talk) 02:15, 5 June 2012 (UTC)

Frankly I don't know what a "rectangular arrangement" is in mathematics. Nijdam (talk) 12:24, 5 June 2012 (UTC)

Since this encyclopedia is written in English, rather than mathematics, that limitation of the language of mathematics is not a problem.TR 13:08, 5 June 2012 (UTC)

That's fine for the introduction, but not for the definition section. Nijdam (talk) 10:28, 6 June 2012 (UTC)

I don't see any major deficiency in the description as it is now. There is no reason to go out of the way to make it more technical. It goes without saying but: WP is not always bound by the same rigors as a textbook. Rschwieb (talk) 18:42, 6 June 2012 (UTC)

Who says it should? But the so called definition is no more than a repetition of the description in the introduction. And in an encyclopaedias also a formal definition should have its place. Nijdam (talk) 12:21, 7 June 2012 (UTC)

Have you got a citation which says something quite similar to what you are saying? Dmcq (talk) 13:57, 7 June 2012 (UTC)

"matrixes" vs "matrices"

Since the blasted editor submitted changes when I pressed the right arrow key (WHY??!!), I was unable to finish my edit summary. I was about to say that googlescholar had the ratio of occurences of "matrixes" to "matrices" at 103k to 2330k. That is, of all those hits "matrixes" accounted for about 4%. My speculation is also that in mathematics, the ratio is even lower. I think "ixes" is frequently even discouraged.

Since usage is this rare, it does not seem appropriate to represent this usage in anything more than a footnote. Rschwieb (talk) 12:53, 27 September 2012 (UTC)

Google books puts the ratio even lower (at 2% = 194000/9740000). I would go further and suggest that this sort of language variant does not warrant any mention whatsoever unless it serves to clarify a possible confusion of interpretation. Such language variations fall entirely into the scope of a dictionary, which Wikipedia is not. Otherwise, for example, we'd have to list "minimums" as an alternative to "minima" as plural of "minimum", which my dictionary does, but Wikipedia doesn't. — Quondum 13:50, 27 September 2012 (UTC)

What's wrong with "minimums" as an alternative to "minima"? It's perfectly legitimate and should be included. Is there a Wikipedia rule that says alternative spellings should be censored out? I don't think so, otherwise http://en.wikipedia.org/wiki/Stadium, http://en.wikipedia.org/wiki/Lemma_(mathematics), should be changed (or footnoted as Rschwieb suggests) as well. Your position forces you to scour Wikipedia searching for "formulas" and replacing with "formulae". The role of Wikipedia is to inform, not to conform, and allowing for all possible spellings, rare as they may be, would help us focus on contents rather than wasting time with form. cerniagigante (talk) 21:34, 23 October 2012 (UTC)

You seem to have missed the point. No-one has suggested any such "forcing". We are simply saying that it is inappropriate to modify the article to emphasize a word usage that is so minor; in a dictionary it would be appropriate to list all accepted plurals. The plural "matrixes" may be used in some articles, and since it is correct, it would not generally make sense to change it in those articles. Usage within an article should be consistent, and it is inappropriate to make sweeping changes. My position is merely that it is inappropriate to introduce a new plural (plus inconsistent usage, as it happens), and was thus reverting it to its earlier consistent usage. — Quondum 13:57, 24 October 2012 (UTC)

I am aware that "matrixes" is much less frequent than "matrices" and did say that in the part that you obliterated (without asking). But, whether you like it or not, and small as it may be, the percentage is not 0. I thought it is Wikipedia's policy it is better to state a fact than not, especially if someone bumps into that minority of articles and books where "matrixes" is used.

@Cerniagigante You also learned nothing from the numerical support given in my argument. Had you applied a similar googlebooks/googlescholar search technique, you would find that "formulae" has 61% of the usage of "formulas" on googlebooks, and 66% of the usage of "formulas" on googlescholar. So, that indicates two things: A. "formulas" is more preferable, and B. "formulae" has strong usage, so argument to replace all occurences of "formulas" with "formulae" is extremely weak.

That should serve as a second proof that relying on your anecdotal feelings has lead you to the wrong conclusions. And I am sorry if I sound gruff, I am just trying to say it plainly. I do hope you adopt some similar reasonable measure before you make edits of this nature, though. Rschwieb (talk) 15:19, 24 October 2012 (UTC)

Thank you for clarifying (it was not needed though, I did note your numbers) but as with all stats you need to agree on a threshold to declare something insignificant. If there is such a threshold, please quote, otherwise your unilateral decision amounts to a point of view. Please note that I am not basing anything on anecdotes: I am quoting from widely used dictionaries such as OED and Webster. On the other hand, by running stats on google, you are conducting lexical research, which is another thing I understand we are not supposed to be doing here on Wikipedia. (Note that you misspelled "occurences" ;-) cerniagigante (talk) 10:43, 7 January 2013 (UTC)

In cases like this google's ngram viewer is your friend. matrices vs. matrixes, minima vs minimumsformulae vs formulas lemmata vs lemmas vs lemmae stadia vs stadiums (the last one has a weird history).TR 15:45, 24 October 2012 (UTC)

Very cool: thanks for the link. Rschwieb (talk) 16:17, 24 October 2012 (UTC)

Nice tool. (For lexical research.) cerniagigante (talk) 10:54, 7 January 2013 (UTC)

@cerniagigante Petty sniping at spelling is not impressing anybody (especially since you are arguing for "matrixes") so I hope we can skip it in the future. Mocking an obvious litmus test for commonality of a word is also a waste of time that does not advance your argument.

Asking about what threshold of use is a better question. Five percent already seems like a very generous threshold. "Not zero" is a patently bad (and absurd: there are no zero usage terms in writing) I wonder if you meant something else. Perhaps you meant that you think 1% or 2% is acceptable?

Another way you could support yourself is if you found two or three reliable secondary resources that support the usage in a mathematical context. Regards, Rschwieb (talk) 20:12, 7 January 2013 (UTC)