Talk:Matrix multiplication

Dot products of vectors

A good way to envisage matrix mult is to split the first into rows, the 2nd into columns, and vector dot-produc them. --anon

Matrix multiplication can also be envisages a dot products of vectors. The above example becomes::

${\begin{bmatrix}\mathbf {a} _{1}\\\mathbf {a} _{2}\end{bmatrix}}*{\begin{bmatrix}\mathbf {b} _{1}&\mathbf {b} _{2}\end{bmatrix}}={\begin{bmatrix}\mathbf {a} _{1}\cdot \mathbf {b} _{1}&\mathbf {a} _{1}\cdot \mathbf {b} _{2}\\\mathbf {a} _{2}\cdot \mathbf {b} _{1}&\mathbf {a} _{2}\cdot \mathbf {b} _{2}\end{bmatrix}}$

The above is for the article, but trying to get the numbers right makes my brain ache. I'm leaving it here in case I've got them wrong. -- Tarquin 17:30 Jan 15, 2003 (UTC)

Perhaps it is good to work with column vectors; then the a's on the left must get a T for transposed. - Patrick 17:39 Jan 15, 2003 (UTC)

yup, that's the right way to do it.

I like that way a lot better than the other way, and I think the above general formula should be on the page. How I usually envision it is I write a horizontal line, then a verticle line, representing a lines of vectors. Its simpler to me to write a matrix of dotted vectors, rather than handle every element of the matrix separately. Fresheneesz 08:35, 21 March 2006 (UTC)

After reading over the "proportions-vectors method" I don't have any idea what it means by proportions.. Perhaps that should be explained better? Fresheneesz 08:37, 21 March 2006 (UTC)

"we take once the first vector and twice the third vector, while ignoring the second vector"

I'm pretty sure we're not gonna "ignore" anything, you multiplied the second vector by 0. I'm going to change that so it doesn't sound as .. vapid.. (no offense). Unless anyone argues. Fresheneesz 08:39, 21 March 2006 (UTC)

Picture

Okay, here's something that is probably one of these things that only makes sense to me.

When this picture was posted, it looked different. Find the original here.

it basically shows that the entries of the product matrix are filled in according to which row and column are multiplied. If anyone else gets it & thinks it useful for the article , please add it -- Tarquin 23:46 Jan 21, 2003 (UTC)

(except I've just realised the result matrix in the pic has the wrong number of rows .... hmmmmm. if there's a call for it I'll remake it)

Surely it makes sense to me also! This diagram uses unequal m and p, which is better (more general). I would use the diagram and add a third column for B in the example. - Patrick 01:14 Jan 22, 2003 (UTC)

Ah, that's good to know! There are many concepts in maths that I envisage pictorially in some way... and then I find that nobody else does... very disconcerting! Well, if one more person gets it, we'll put it in the article :-) -- Tarquin 23:03 Mar 14, 2003 (UTC)

This is the best illustration of matrix multiplication I've ever seen. In fact, I think it's one of the best math illustrations on Wikipedia. The second I saw it, I 'got' matrix multiplication for the first time. Beautiful work. Fredrik | talk 23:24, 10 Mar 2005 (UTC)

I fixed the picture, making it have correct number of rows. I also added yellow highlight around the elements that are being used, cause before it could have looked like only two elements were going into the new cell (the cells from which the arrow starts). Thats how I saw the picture so it confused me. I think the outline helps clarify that confusion. Fresheneesz 09:47, 21 March 2006 (UTC)

It looks like you've messed up the description. In the matrix operation AB, the column count from A and the row count from B are taken to form the resulting matrix AB. In other words, the operation shown in your picture is BA, not AB. —The preceding unsigned comment was added by CodeMercenary (talk • contribs) 07:35, 9 April 2007 (UTC).

I created a modified version of the image on the page. I added the names for the items in the matrix (a_1,2 etc) to make it clearer how the multiplication works. Do you agree that this image is better? Would like comments before inserting it. Lakeworks (talk) 23:03, 14 January 2008 (UTC)

Two minor suggestions (which are a matter of taste):

1. Make the cells square, and mark the two targets with non-flattened circles.
2. Also fill in the "unused" cells of A and B: a_2,1 etcetera, using a grey colour.

 --Lambiam 06:45, 15 January 2008 (UTC)

Updated the image with the suggestions you made. Any more suggestions or can I update the image in the article?

Lakeworks (talk) 21:11, 16 January 2008 (UTC)

It's fine with me (but I can't speak for other editors).  --Lambiam 01:47, 17 January 2008 (UTC)

Go for it. -- Jitse Niesen (talk) 15:12, 17 January 2008 (UTC)

This image is very helpful for me to refresh my memory, thanks for the addition Lakeworks. James Lednik (talk) 00:21, 14 February 2008 (UTC)

This image does not match the example. The example results in a 2x2 matrix and the example results in a 3x3. Would it be easier to understand if they matched? —Preceding unsigned comment added by 128.112.151.175 (talk) 02:01, 7 April 2009 (UTC)

Directly next to the picture it says: "For example BA:" and lists the BA multiplication. That is confusing, as the picture shows the AB multiplication. I agree that BA should also be shown, but first show the AB numbers that go with the picture! --96.234.240.247 (talk) 13:11, 24 April 2009 (UTC)

Lakeworks's version of the image better illustrated the concept, in my opinion. It showed unequal dimensions, i.e. with A mxn and B nxp, C is mxp. Could we restore it? --StefanVanDerWalt (talk) 14:07, 4 October 2010 (UTC)

HTML representation

While I love HTML, the statement:

(the HTML entity ⊗ (⊗) represents the direct product, but is not supported on older browsers)

seems out of place. This article is about matrix multiplication, not HTML. Thoughts?

I've seen another article which had a simliar comment, and the consensus was to leave it because "people deserve to know how to write it" or something. I think its small, and should be at the end, but be there none the less. Fresheneesz 08:35, 21 March 2006 (UTC)

Thank you! I cannot imagine things pictorally well, and I always forget the algorithm for Matrix Multiplication. This picture is a lifesaver! wiki@matthewwilkes.name

Error in Kronecker Product Section

In the Kronecker product section, I believe there is an error.

${\begin{bmatrix}a_{11}B&a_{12}B&\cdots &a_{1n}B\\\vdots &\vdots &\ddots &\vdots \\a_{n1}B&a_{n2}B&\cdots &a_{mn}B\end{bmatrix}}$

Should be:

${\begin{bmatrix}a_{11}B&a_{12}B&\cdots &a_{1n}B\\\vdots &\vdots &\ddots &\vdots \\a_{m1}B&a_{m2}B&\cdots &a_{mn}B\end{bmatrix}}$

Right??

Yup. There should be an m. Fixed it. -- Tarquin 18:14, 22 Jun 2005 (UTC)

partitioned matrices

There needs to be some explanation of Partitioned matrix algrbra especially with respect to multiplication. I don' the math software so I hereby throw the ball to someone else. MPS 14:53, 20 Jun 2005 (UTC)

Notation

It is quite unusual to write AxB for the matrix product. AB or if necessary A.B is commonly used. Also, for multiplying numbers one uses 3x4 and not 3.4. The dot is used for the product of variables: a.b if just writing ab would lead to confusion.Nijdam 23:14, 28 February 2006 (UTC)

I agree that A × B for the matrix product is very uncommon, and I edited the article accordingly. However, I've encountered the notation 2 · 3 for the product of the numbers 2 and 3 quite often, so I left that one in. -- Jitse Niesen (talk) 11:20, 2 March 2006 (UTC)

Yeah, straight juxstaposition is probably the most common thing.... but I have another observation. I think that the subscripting is quite clear in general, but separating indices with a comma conflicts with the use of commas before an index in general (virtually ubiquitous) tensor notation as a means of indicating differentiation with respect to the index following the comma. I think the article is so clear and useful that this should not be taken as a severe criticism and I am not motivated to "boldly" change this. Perhaps a brief discussion about whether this should be attended to is a better idea. scanyon 01:42, 20 February 2010 (UTC) —Preceding unsigned comment added by Patamia (talk • contribs)

howto's belong in wikibooks

wikipedia should just say what matrix multiplication is

I agree. This is a how to, and doesn't belong in an encyclopedia any more than the receipe for goat stew.

I disagree. This is exactly the information I was looking for when I searched for Matrix Multiplication. This article does not represent a how to, only an appropriately rigorous explanation of what matrix multiplication is.

I also think that this article is both fine in general and extremely useful even in an encyclopedia context. How an "operation" is defined often entails explaining "how" the operation is performed. Beyond that there is a "matter of taste" issue, but in this particular case, I vote to certify this article as more than worthy to remain in the encyclopedia. I have a reasonably broad and in several respects advanced background, but when I want to remind myself of what something IS I become exceedingly dismayed when I encounter a Wikipedia article that is so esoteric and so dependent on foreknowledge of advanced notation that is is effectively useless. I understand that there is a justifiable concern about whether an encyclopedia article should be pedagogical, but for some fairly fundamental mathematical definitions I think that explaining what something is becomes indistinguishable from explaining how it works.scanyon 01:36, 20 February 2010 (UTC) —Preceding unsigned comment added by Patamia (talk • contribs)

I too agree that this is an excellent and relevant article and should be kept as-is. 10:42, 24 January 2011 (UTC) — Preceding unsigned comment added by 51kwad (talk • contribs)

The coefficients-vectors method

In the example, it jumps from [3 1]+[0 0]+[2 0] to [5 1] without any explanation. I've had every edit I've made to a math page reverted, so I will suggest here that an extra step be added to point out how you get from the first step to the next. It is [(3+0+2) (1+0+0)]. Without that, it is not obvious what is going on unless you already know matrix multiplication - and if you did, you wouldn't be reading this article. --Kainaw ^(talk) 14:57, 2 September 2006 (UTC)

I think it's pretty obvious as it is. The [a b] is notation for a vector, so k[a b] = [ka kb] and [a b] + [c d] = [a+c b+d] (by linearity). Adding in the additional step does clutter things up a bit. —Preceding unsigned comment added by 129.31.68.254 (talk) 00:05, 1 September 2008 (UTC)

proportions matrix?

In "properties, what the proportions matrix means?

The article either needs to explain or avoid this usage. The "proportions" matrix is the left-hand matrix and the "vector" matrix is the right-hand matrix. Paul D. Anderson 23:13, 6 October 2006 (UTC)

Inner product

Is there a word for taking the inner product of two matrices as though they are vectors? That is

$x=\sum _{i,j}M_{ij}M_{ij}=M:M=M_{11}+M_{12}+\cdots +M_{21}+M_{22}+\cdots$

So, is there a name for this? —Ben FrantzDale 17:03, 20 December 2006 (UTC)

I'm not completely sure what you mean, but I think you're talking about the "Frobenius inner product". For two m-by-n matrices A and B, that is defined by

$\langle A,B\rangle _{F}=\operatorname {trace} (AB^{\top })=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}.$

You need to take conjagates for complex matrices. This inner product leads to the Frobenius norm, but that seems to be the only reference on Wikipedia to the Frobenius inner product. -- Jitse Niesen (talk) 18:16, 20 December 2006 (UTC)

That's the one. Thanks. Do you think it would be appropriate to define Frobenius inner product on this page? —Ben FrantzDale 21:14, 20 December 2006 (UTC)

Yes, it could be mentioned. Perhaps another page would be more appropriate, but I'm not sure which one, so just add it here. -- Jitse Niesen (talk) 16:47, 21 December 2006 (UTC)

Will do. It looks like this is closely related to Frobenius algebra. I don't know enough of the details to be able to say how these topics should be divvied up among pages, though. —Ben FrantzDale 18:03, 21 December 2006 (UTC)

This article sucks

I nominate that this article gets the suck award. 136.159.197.83 20:44, 17 April 2007 (UTC)

Totally disagree. This is an excellent article, and should stay. 51kwad (talk) 10:44, 24 January 2011 (UTC)

I'm lost

What is r, is this rows? Please define the variables in the example before using them. I'm looking somewhere else for something easier to follow. —Preceding unsigned comment added by 72.230.164.82 (talk) 18:32, 11 October 2008 (UTC)

Article needs a good simple illustration

Maybe something like the Generalized Example here http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php —Preceding unsigned comment added by 213.104.208.37 (talk) 15:50, 3 December 2010 (UTC)

way too complicated

seriously how is the average joe meant to understand this? please could someone simplify it

I agree, I just wanted to see how to simply multiply 2 matrices; and multipy a scalar and a matrix, but this is site didn't help. —Preceding unsigned comment added by 129.21.40.143 (talk) 22:15, 11 February 2008 (UTC)

You are right. This page could do with a simple introduction. In my junior-high curriculum, we ran into matrix multiplication in 8th grade (age 12ish), well before I ever saw summation notation and well before we saw any applications for matrices. I just added a little bit, but more could be added to make this article much more accessible. While it's easy to state the algorithm for matrix–matrix multiplication, I find it impossible to understand before you have an intuition for what the vectors themselves mean. Then matrix–vector multiplication is just a way to make a new vector by having each entry in the new vector be some weighted sum in the initial vector. Then matrix–matrix multiplication is the "right" way to find P=BA such that Px=BAx. I hope that helps and I hope people work to make this article more approachable. —Ben FrantzDale (talk) 01:33, 12 February 2008 (UTC)

What the article needs is some examples for multiplying simple matrices (e.g. 2x2 and 3x3). People who aren't familiar with the other notation will most likely only need that. Andrewy (talk) 02:08, 8 December 2008 (UTC)

Why would the average joe want to read this article? And why should an encyclodedia be designed for the average joe? The proper context for introducing the matrix product is Linear Transformations of Vector Spaces and Modules, but that would be beyond most readers. However, it is good to have technical articles in WP. 51kwad (talk) 10:52, 24 January 2011 (UTC)

Broken link?

The link for "Efficient Matrix Multiplication for single dimension arrays" (http://angelo.freeshell.org/computer/docs/mmult.php) appears to be down (503 error). Is there another reference to that article? A google search didn't find it. -- Ed Burnette

An copy archived on 10 May 2007 is available at http://web.archive.org/web/20070510025901/http://angelo.freeshell.org/computer/docs/mmult.php . However, I don't think it is a useful resource, so I removed the link. -- Jitse Niesen (talk) 04:39, 2 August 2007 (UTC)

Possibility of adding a section explaining why matrices work the way they do

Does anyone else think that it wouldn't harm to mention how matrices can be seen as maps between vector spaces and that matrix multiplication corresponds to the composition of maps (after picking a basis)? LkNsngth (talk) 03:50, 3 May 2008 (UTC)

It is important to add that. Matrix multiplication is defined in such a seemingly non natural way because it corresponds precisely to composition of associated linear transformations. A new section might be in order.--Shahab (talk) 04:48, 3 May 2008 (UTC)

I've started working on such a section.  --Lambiam 16:34, 3 May 2008 (UTC)

 Done; please review and improve.  --Lambiam 18:07, 3 May 2008 (UTC)

I believe there was a small technical error in your writing, a vector space is constructed over a field. Modules are generalizations of vector spaces over a ring. Other than that it looks good LkNsngth (talk) 23:18, 5 May 2008 (UTC)

That means that matrices over integers are excluded, so I removed the reference to integers. -- Jitse Niesen (talk) 08:38, 6 May 2008 (UTC)

Powers of matrices

Powers of square matrices are a special case of matrix multiplication, so it seems to me they should be at least briefly discussed in this article. I'm going to add a brief section on this topic; please remove if there's a problem and I'll help improve it or find some other place to put it, but I really think there needs to be some coverage of this topic on Wikipedia. Swift1337 (talk) 05:47, 16 October 2008 (UTC)

OK, I've added it. It's pretty barebones and could use some discussion of convergence of powers of matrices, but I think it'll be useful as a starting point at least. Swift1337 (talk) 06:07, 16 October 2008 (UTC)

Change example matrix

The example matrix is a bad one because the outer dimensions are the same too. This might give the false impression that the matrices have to have rotated dimensions. I would recommend adding a row to the first one, making it the same size as the picture. Asmeurer (talk ♬ contribs) 17:21, 15 November 2008 (UTC)

I agree that this example would be improved if the matrices had different outer dims. BeckyAn (talk) 01:04, 7 December 2008 (UTC)

Should coalesce two properties sections

There are two "properties" sections -- Properties in the upper-third of the page, and "Common Properties" tucked down amongst the different specialized products. I propose consolidating them, into the earlier of the two sections. BeckyAn (talk) 01:04, 7 December 2008 (UTC)

premultiplication and postmultiplication

If I make the product AX, I premultiply X by A. If I make the product XA, I postmultiply X by A.

Is this the right usage of premultiplication and postmultiplication? Are these words authorized terms or slangs?

I'm a stranger in English. Bmack7264 (talk) 09:10, 3 June 2009 (UTC)

I'm no native either, but I think it is more common to say left and right multiplication. Jakob.scholbach (talk) 11:09, 3 June 2009 (UTC)

Clean-up

This article is a hodge-podge of different topics, and I feel that it would be worthwhile to restrict the scope to just the standard matrix product. Majority of the other topics have their own dedicated "main" articles.

I've begun the clean-up by correcting some errors, streamlining and rearranging the material in the most important section on the matrix product. I removed a lengthy and non-informative subsection illustrating the composition, which also violated WP:NOTTEXTBOOK. Several other sections run afoul of the same policy, in my opinion. Weighted matrix product is potential OR and is next in line for deletion, unless a serious case is made for preserving it, possibly as a separate article. Also, I am less than enamoured by non-standard notation in the subsection illustrating the matrix multiplication, which sports sexy different colors, but mixes latin characters and arabic numerals in referencing matrix entries. Certainly, much work remains to be done. Arcfrk (talk) 06:04, 2 September 2010 (UTC)

Thanks for calling attention to this. The article should, of course describe how matrices are multiplied, but giving several explanations with multiple examples crosses into HOWTO/TEXTBOOK territory. I'm thinking that some of this material should be merged in Wikibooks instead of deleted; with a link from this article people who are looking for a textbook style treatment will be able to find it easily. I did find a couple of brief mentions in a Google search on weighted multiplication, so I don't think it's entirely OR, but I doubt there is enough material out there for it to be encyclopedic and I'm sure major parts of the section are OR. So I'd agree that the section should be deleted unless significant improvements are made and references added. I agree also that the sections which describe other types of multiplication have content fork issues and should be removed or at least shortened to a single sentence. This would be in keeping with summary style, but besides that I think these other types of multiplication are separate subjects and don't really belong here. I'm all for collecting similar subjects together to make a half-decent article from a collection of stubs, but to me matrix multiplication is really a way to compute the composition of linear maps and these other forms of multiplication have little to do with that. I'd also note that the lead section does not seem to be in line with the MOS and could probably use a rewrite.--RDBury (talk) 08:24, 2 September 2010 (UTC)

The "The Weighted Matrix Product" section keeps being re-inserted by the same user, User:Cloudmichael. As far as I can tell, the section is pure WP:OR. A comment[1] left by User:Cloudmichael at User talk:Sławomir Biały (after Sławomir Biały removed that section), indicates that User:Cloudmichael does not understand that Wikipedia does not publish original research. Nsk92 (talk) 11:50, 5 September 2010 (UTC)

Product

I'm not sure whether every matrix necessarily has to be considered as the representation of a linear map. That's why I didn't refer to this in the introduction. Nijdam (talk) 09:54, 5 September 2010 (UTC)

Too technical?

This is as easy as it gets! Very good article IMHO. I propose the banner be removed? 51kwad (talk) 10:39, 24 January 2011 (UTC)

I believe I put the banner up. The article is currently rated C class which is two grades removed from good. The second sentence of the lead doesn't help anyone who is not already familiar with the advanced concepts linked there. There are two intimidating equations in the first section. The goal is to make it understandable to non-experts (e.g. someone trying to get help with their math homework). The Illustration section would be better place to start than "composition of the linear transforms". Let me know if further (hopefully) constructive criticism is needed. --Kvng (talk) 19:25, 24 January 2011 (UTC)

I guess it arises from trying to teach matrices before people have learnt the basics of groups, rings, fields and vector spaces. That is the proper context. From years of teaching this naively, the only way of justifying matrix multiplication seemed to be in terms of substituting one set of linear equations into another. 51kwad (talk) 13:29, 27 January 2011 (UTC)

The article is definitely focused upon too narrow an audience. It's not that it's too technical, it's that it lacks a basic explanation of what matrices are and what they might be used for. The article also meanders around to topics that really don't belong here, such as various algorithms for computational efficiency. Obviously such algorithms should have there own articles and do nothing to explain what matrices are.

It's almost humorous that the article is so long and goes into such depth about how to compute matrices, but completely misses why you might want to do so. Gatohaus (talk) 14:55, 1 February 2011 (UTC)

Er… Which article are you commenting on? Obviously, the explanation of what matrices are belongs to the article matrix (mathematics), and not the article on matrix multiplication; an explanation of why you might want to multiply matrices occurs in the second sentence of the lead, and then again in the second sentence of the first section. Where I agree with you is that algorithms for efficient computation of the matrix product should eventually be spawned off. Arcfrk (talk) 01:32, 2 February 2011 (UTC)

Thinking the low grade and overtechnicality could be solved if matrix multiplication could have a disambiguation page and separate, better articles for different depths of the material. One article for multiplication of matrices with real values with applications to linear algebra, and another one with abstract algebra considerations and more advanced concepts. Matrix multiplication is used in different contexts. Formally it is not ambiguous but in human understanding it is. Wikivek (talk) 21:11, 9 February 2011 (UTC)

No, I disagree. There is no need to write a second article (and doing so would actually harm both resulting articles). Instead, this article here should start out with a section taking two concrete matrices (with integer entries, say) and calculating their products. It is also conceivable to make an example of multiplication of two 2-by-2 matrices, and relating the product to the composition of the corresponding linear maps. Jakob.scholbach (talk) 21:23, 9 February 2011 (UTC)

I also oppose splitting the article. The article needs to be more accessible especially at the beginning. The article could be improved by bringing in some of the text from the matrix article. --Kvng (talk) 15:25, 11 February 2011 (UTC)

The article is indeed too technical. Let's simplify a step at a time, for example, the second sentence of the summary is already covered in the content and doesn't add anything to summarize what the article is about. It will confuse anyone if they jump to those topics which are way beyond matrix multiplication. Also, the first part of the content shouldn't be implying that the reader jump to topics on Linear Transforms or vector spaces...it should give the reader some insight into what matrix multiplication does...not a technical reason for why it's possible.