Talk:Matrix (mathematics)

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[edit] Inverses of Matrices

There appears to be nothing on Wikipedia about finding the inverse of a matrix manually -- I think this is a pretty major omission, and I only started the module of my course on matrices last week. There is some sparse mention of uses of a matrix inverse, but the methods for finding inverses and, I believe, determinants are lacking, or just hard to find.

Perhaps someone will advise me, otherwise I shall upload some stuff on finding inverses and determinants on sunday [probably]. EdJ343 15:28, 14 March 2007 (UTC)

See Invertible matrix. In the future please add new comments at the end of the talk page.MathMartin 15:50, 14 March 2007 (UTC)
If you go to the section Square matrices and related definitions in the article, you'll see all these concepts shortly mentioned. There are links to invertible matrix (as MathMartin said) and determinant where you can find more information. -- Jitse Niesen (talk) 00:45, 15 March 2007 (UTC)
Thankyou, my mistake. I have learnt something. EdJ343 07:20, 19 March 2007 (UTC)

[edit] Notation issues

What does: "The notation A = (aij) means that A[i,j] = aij for all indices i and j. " mean? What is a, is it another matrix, or a contant or what? -- SGBailey 22:15 Jan 17, 2003 (UTC)

There is no a, just a11, a32, etc. -- Wshun Jan 21

There are several ways of notating the (i,j)th element. A[i,j] is one; aij is another which is easier on the eye. the small a is used to emphasize that it is a number. Also, Aij is used for the matrix A with some sort of manipulation to the (i,j)th element or ith row & jth column. This probably needs adfding to the article -- Tarquin 23:19 Jan 21, 2003 (UTC)

You mean I could rephrase the original quote as:

" The notation A = (A[i,j]) means that A[i,j] = A[i,j] for all indices i and j. " -- It seems overcomplicated to introduce an alternative set of nomenclature for this "one page" article. I suggest we either stick to one method throughout to explain matrices or we consider both nomenclatures important enough to be explained as part of the article and explain them and give an example in each case. -- SGBailey 23:32 Jan 21, 2003 (UTC)

[edit] More notation issues

I'm confused about notation here. What's with the (parentheses) and [brackets]? When do we use one notation, and when the other? What's the difference, if any, between (aij) and [aij]? MathWorld uses the same notation, but doesn't explain well either.
—Herbee 01:05, 2004 Feb 26 (UTC)

The notation here seems consistent: for example with a vector it's a[i] for i-th component of the vector, and (ai) for the whole vector written as a list of indexed numbers.
Charles Matthews 09:01, 26 Feb 2004 (UTC)
Yes, I see it now—thanks for kicking my eyes open. I was looking for a deeper meaning in a badly designed page...
Wikipedia turns out to be inconsistent on matrix notation, so there is little point in fixing this one page. We should really convert everything to standard mathematical notation. I might even volunteer, except that I wouldn't know how to track down all the relevant pages. Anyone?
—Herbee 12:50, 2004 Feb 26 (UTC)
You could take that to Wikipedia talk:WikiProject Mathematics. Paradoxically (or perhaps not) the maths here grows apace, but the standardisation of how it's written is pretty much neglected.

Charles Matthews 13:43, 26 Feb 2004 (UTC)

[edit] equivalence relations

Could someone do a section on the different equivalence relations that are defined on matrices? There's similarity, but I'm sure I remember one that worked with the transpose.

The latter is in relation to bilinear forms, where ATMA can replace M by change of basis. I forget the name for it.

Charles Matthews 09:03, 26 Feb 2004 (UTC)

[edit] Matrix multiplication

Perhaps there should be some explanation of why multiplication works the way it does? It seems somewhat arbitrary to me.

Historically it was certainly discovered in relation to choosing new variables in simultaneous linear equations. These days we'd probably say that it is a question of having matrix multiplication match up with composition of linear transformations.

Charles Matthews 09:47, 28 Jan 2004 (UTC)

I see it particularly obvious when you start by writing a system of 3 equations and 3 variables so that each equation has its terms ordered (say, first the x term, then the y term and last the z term) and terms are aligned vertically. And then you just "factor out" the variables as a vector which multiplies the coefficients by the right. Well, an image is worth a thousand words:
 1 x  + 5 y  - 1 z   =   9
 2 x  - 4 y  + 1 z   =   9
-3 x  + 3 y  - 1 z   =   9

Becomes:

 1    + 5    - 1       x    =   9
 2    - 4    + 1   X   y    =   9
-3    + 3    - 1       z    =   9

If you wonder about matrix by matrix, and not matrix by vector, well, it's just like forming the right matrix and the result matrix by juxtaposing column-vectors. --euyyn 19:17, 3 February 2007 (UTC)

[edit] rotation matrix

==3D-Rotation of any vector (x,y,z) around an axis of the
  direction(a,b,c) by an angel @== 
We reduce the vector of the axis-direction to the length 1:
(1/ sqrt(a^2+b^2+c^2))* (a,b,c)=(A,B,C).
Reckon the following and you get the result of the rotation
    1  0  0             0  -C   B                  0   -C   B   2       x
[ ( 0  1  0 ) + sin@* ( C   0   -A ) +(1- cos@)* ( C   0  -A )    ] * ( y )
    0  0  1             -B  A    0                -B   A   0            z
(Notice, that the third matrix must be squared and then multiplied by cos@)
Imagine a plane, to which the axis is normal to and in which lies the tip of
the arrow (that is the picture of the vector)In this plane you add an arrow
from the tip in the direction of travel -that is the orientation of the rotation.
And from this you add another one in this plane in the direction of 90 degrees
to the left respective to the previous one.
The vector (x,y,z) and the result of the formula above are of same length.
The angle between these two is not the angle of rotation - the tip of the arrow
is rotated in the plane , which is perpendicular to the axis.

==Extract axis and angle out of a rotation-matrix==
A rotation-matrix D has the property: det D =1 and D * D(T) = E , where D(T)
is the matrix transponed, that is you interchanged colums and rows and
E is the unit-matrix.
A matrix can be split into a symmetrical and an antisymmetrical

(a(ik)= a(ki) ) and (a(ik) = - a(ki) )

So:
  a d e           2a   d+g    e+h           0     d-g    e-h
( g b f ) =1/2*( d+g    2b    f+i ) +1/2*(g-d      0     f-i )
  h i c          e+h   f+i     2c         h-e     i-f     0
The antisymmetrical part gives the direction af the axis: (i-f, e-h, g-d )*1/2.
The length of this is sin@.
The main diagonal of the matrix gives the "spur": a+b+c and this equals
1 + 2*cos@. From these you get @.
An extra-bonus: The affin mappings (if this is the right word), that is here
the 3*3-matrices can be split in an symmetrical and an antisymmetrical
part. The first you explore by means of main-axis transformation and the
antisymmetrical ones - applied to a vector - correspond to the
cross-product:
  0   -c    b       x
( c    0   -a ) * ( y ) = (a, b, c ) x (x, y, z )
  -b   a    0       z
Hero van Jindelt

[edit] Block diagonal matrices

Block diagonal matrices / diagonal block matrices: should there be a seperate entry for this type of matrix, or could it be added to diagonal matrix? Chris Wood 20:09, 9 Mar 2004 (UTC)

[edit] In the sequel

Under the category of "Linear transformations, ranks and transpose," the second paragraph begins "Here and in the sequel we identify..." In the sequel?SWAdair | Talk 11:46, 24 Mar 2004 (UTC)

[edit] Jargon

This page is full of words that someone that doesn't remember this stuff from math class or never learned it would not understand...and my math textbook explains a lot of this stuff a lot more clearly than this page does. argh. Some changes need to be made, but I'm not sure how to go about that. — Braaropolis | Talk 00:13, 28 Jun 2004 (UTC)

  • Well, I disagree that it necessarily needs to be changed a great deal, since it would be pointless to only include the information that the "average" person knows about matrices (which is pretty close to zero, in my opinion). I understand pretty much everything on this page, and well I should, but I think it should stay pretty much as is. Quandaryus 19:38, 5 Sep 2004 (UTC)

[edit] Refactoring of article

I agree with User:Braaropolis the article is in bad shape. It is too long and the scope is to wide. The basic article on matrices should be as accessible as possible as the topic is so central to linear algebra. I tried reordering the material to make it clearer and moved the content of Partitioning matrices to block matrix. But the article is still too long. Perhaps we should put square matrices into a separate article and move some topics of the matrix atricle into matrix theory (in the same way graph_(mathematics) is related to graph theory) MathMartin 15:09, 26 Sep 2004 (UTC)

[edit] Rings vs. semirings as foundation

The current revision states that the entries of a matrix are generally elements of a ring. This is too specific. Matrix addition and multiplication, as defined here, do not require additive inverses. In fact, these definitions apply unchanged if the underlying algebraic structure R is a semiring. This is of crucial importance in graph theory and formal language theory, since e.g. the algebraic structures underlying weighted graphs can often be arbitrary semirings and do not have to be rings (for example, Kleene algebras). I know this is getting far afield, but the generality of matrices over semirings is essential in many cases, and the distinction of matrices over rings vs. matrices over semirings is often crucial. For example, all sub-cubic-time algorithms for matrix multiplication I'm aware of assume at least matrices over rings and do not generally apply to matrices over semirings. --MarkSweep 07:56, 30 Sep 2004 (UTC)

[edit] reordering sections

Can we move Matrices with entries in arbitrary rings to the bottom since it's more abstruse then the rest?

Since history has only one entry (the link to matrix theory) can we incorporate it into something near the beginning? RJFJR 16:32, Dec 24, 2004 (UTC)

I moved the history. First, the way it was before, in front of the definition, was inappropriate (you don't get to writing history of things before you define them!). Maybe the history can be moved up, but where? Maybe after Examples, because the sections below it fit together very well. On the other hand, I think in a math article the history should be the last entry. Not because history is not important, but because in math the properties of things are more important than history.
I want to mention that the article Matrix theory advertized as "Main article" is a very poor article. It has no theory, only history, and elementary introduction. I would suggest the history be moved back to the main page, the elementary introction too, and then, having all stuff in one place, do lots of thinking of how to organize things better, because the way things are now is not good. Too much stuff!
I agree with you, the entry Matrices with entries in arbitrary rings needs to go towards the bottom. This section is not as elementary as others.
Looking forward to feedback! --Oleg Alexandrov 01:56, 25 Dec 2004 (UTC)


[edit] about the definition

The definition currently says that a matrix is a rectangular array of numbers. I'm not sure that's accurate. Isn't a matrix an abstract concept with certain properties? And we represent a matrix as a rectangular array of number? (Am I arguing about what the meaning of 'IS' is? :) )

For that matter can a matrix have something else as a value? In a partition matrix do we have a matrix that is a rectangular array of matrices? RJFJR 00:28, Dec 27, 2004 (UTC)


I think the current definition is fine the way it is. Making things more abstract will make things more confusing for the general public, and this is not what we want.
Yes, a matrix can have anything as value. But again, let's keep things simple. Oleg Alexandrov 02:01, 27 Dec 2004 (UTC)

[edit] removed

The material: Matrix storage uses two conventions: row major and column major ordering. The former means that the matrix is stored such that row elements are packed together contiguously, the latter means that the matrix is stored such that column elements are packed together contiguously.

Belongs in array. It does not refer to the mathematical nature of a matrix but rather to how the values of a matrix are stored in a computer. I am removing it from this article. RJFJR 04:25, Dec 31, 2004 (UTC)

Agree! I also thought that was suspicios (and poorly explained in addition). Oleg Alexandrov 05:26, 31 Dec 2004 (UTC)


[edit] Matrix (mathematics) vs. matrix theory

As of now, there exist two articles on matrices in mathematics, namely Matrix (mathematics) and matrix theory. There is some overlap between them, the logic of splitting the article into two is not clear, and the article matrix theory is very badly written. I suggest that the article Matrix (mathematics) be introductory, listing the definition, examples, and basic properties. The article matrix theory could be the more abstract one. So I think some of the materials of these sections need to be interchanged. I will think more on this. Feedback welcome. Oleg Alexandrov 23:06, 31 Dec 2004 (UTC)


I interpreted it as 'matrix' is the noun and 'matrix theory' is what to do with a matrix. RJFJR 04:06, Jan 7, 2005 (UTC)

I replied on the Talk:Matrix theory page. Oleg Alexandrov 04:36, 7 Jan 2005 (UTC)

[edit] template:matrices

template:matrices - for some cohesion among terms. -SV|t 15:27, 27 Apr 2005 (UTC)

[edit] A section on encrypting

Shouldn't there be a section here on encrypting, since that is one use of Matrices?

[edit] On Applications

You Make no mention of the uses of matrices in computers

[edit] Viewing problem

The problem, somehow, has been solved. Thank you. 203.91.132.17 09:50, 8 November 2006 (UTC)

[edit] Error in example

The first example is said to be a 4*3 matrix, but by the above definition, where it states it should be row*column, it is a 3*4 matrix. It's been like that for a long time. 212.108.17.165 10:14, 27 November 2006 (UTC)

Huh? The matrix has 4 rows and 3 columns, so it is a 4-by-3 matrix. -- Jitse Niesen (talk) 11:45, 27 November 2006 (UTC)

[edit] On history

What was the contribution of the developers of Quantum Mechanics to Matrices? I've read books that claim it was like their invention, but, as explained in the history section here, it's obviously not... --euyyn 19:19, 3 February 2007 (UTC)

[edit] Further reading...

The introductory paragraph mentions the use of matrices with elements from rings. What's a good reference for further reading? --HappyCamper 00:35, 15 April 2007 (UTC)

[edit] Prehistory

Were magic squares really around in prehistoric times, or did someone just get carried away with their terms? Does anyone have a reference? 71.204.151.203 02:49, 15 May 2007 (UTC)

It seems rather unlikely. Prehistory is characterized by an absence of written records, so it's not clear how we would know that magic squares were around at that time. Anyway, there is no reference. I replaced it by something copied from magic square. Thanks for your comment. -- Jitse Niesen (talk) 04:16, 15 May 2007 (UTC)

[edit] Still more notational woes!

All seemed to be going well until I ran into the following:

Here and in the sequel we identify Rn with the set of "columns" or n-by-1 matrices. For every linear map f : Rn → Rm ....


This is another example of the problem of mathematical notation. Rn would normally imply that n is the exponent of R, but above it's given a new, overloaded definition. The "identify with" usage would be more clear if it was changed to "define as". The word "with" implies aggregation, which is not what the author intended.

What exactly is a "linear map", and what does f : mean? What does that arrow mean?

All the math on Wikipedia should be translated into C++. I doubt that anybody would try to overload pow(R, n) to mean "the set of n-by-1 matrices" so the math would become much more readable. true true ....... i might even say a C+.....

216.23.105.3 09:34, 23 May 2007 (UTC)

[edit] `m-by-n' or `m-cross-n'

I used to read and pronounce mxn as `m-cross-n'. Is `m-by-n' commonly used in math community? Shouldn't we make a note that it can be read and pronounced as `m-cross-n' also? 59.178.126.93 17:20, 18 July 2007 (UTC)

For what it's worth, my linear algebra textbooks and professor say by not cross. The former is also consistent with non-mathematical usage (we say two-by-three table not *two-cross-three table). 208.106.1.215 (talk) 18:28, 24 December 2007 (UTC)

[edit] Oh my darling

The tune to the ballad Oh My Darling, Clementine is frequently adapted by secondary school teachers worldwide to teach about matrix multiplication:

Row by column, row by column
Multiply them line by line
Add the products, form a matrix
Now you're doing it just fine.

The authorship of this version has been disputed, but is most frequently attributed to the mathematician/musician Aaron B. Barnett, who published this teaching tool in his seminal work, It's Reciprocal.

The text above the line was recently added to the article. There is no reference, I have never heard about this seminal work and I couldn't find anything about it, so I'm moving it here. Does somebody know a reference confirming this? -- Jitse Niesen (talk) 02:06, 9 August 2007 (UTC)

[edit] Problem Viewing

Is there an editor of this article who can make the script especially on this part, under sum who can make it easier to read (i.e. A + B = (ai,j)1≤i≤m;1≤j≤n + (bi,j)1≤i≤m;1≤j≤n = (ai,j + bi,j)1≤i≤m;1≤j≤n ). As you can see it is readable when you copy and paste but in the article very hard to read, Thanks BigDunc 17:58, 18 September 2007 (UTC)

[edit] Notation for Matrix Addition (Sum)

In my opinoin, the notation at the explanation for matrix addition is incorrect. The first line of

\begin{align} \mathbf{A}+\mathbf{B} &= (a_{i,j})_{1\le i \le m;\, 1\le j \le n} + (b_{i,j})_{1\le i \le m;\, 1\le j \le n}\\ &= (a_{i,j}+b_{i,j})_{1\le i \le m; 1\le j \le n}\\ \end{align}

could (or maybe should) be interpreted as "take any ai,j, where i is between 1 and m and j between 1 and m, and add it to any bk,l, where again k is between 1 and m and l between 1 and m" - since i and j appear twice in different parts of equation, and are free variables in neither of them, they can be renamed.

Therefore, I believe that the first line should be deleted, and only the second should remain. The second line binds both i-s and j-s at the same time, so they cannot be renamed independently.

If nobody answers in 3 days, I will edit the article.

Tom Primožič —Preceding unsigned comment added by 193.77.126.73 (talk) 09:52, 22 December 2007 (UTC)

The definition of addition is given in the second equality:
(a_{i,j})_{1\le i \le m;\, 1\le j \le n} + (b_{i,j})_{1\le i \le m;\, 1\le j \le n} = (a_{i,j}+b_{i,j})_{1\le i \le m; 1\le j \le n}.
The first line of the definition as you quoted above is notation-setting. It just names the entries of matrices A and B. Notice in particular that it does not name the entries of the matrix A + B; that is the job of the right-hand side of the definition. Michael Slone (talk) 12:08, 22 December 2007 (UTC)
Oh, I get it. Thank you for your explanation! Tom Primožič —Preceding unsigned comment added by 193.77.126.73 (talk) 10:24, 26 December 2007 (UTC)

[edit] Generalization to more than 2 dimensions

In programming we would use say a 'three-dimensional array' as a sort of matrix with three axes rather than the standard two considered in this article. Such a matrix could be used to hold some values from a three-dimensional space for example. The article on matrices does not seem to make any reference to any kind of extension from two dimensions to three or more, but surely some results regarding matrices would apply to similar structures with more than just two dimensions. And of course, a one-dimensional 'matrix' would be the same thing as a vector. It seems like a good idea to emphasize common features rather than ignore them, to promote higher levels of understanding, and economy of conceptualization. —Preceding unsigned comment added by 220.253.113.241 (talk) 13:47, 13 March 2008 (UTC)

The article you are looking for is Tensor. Though I agree, it seems a bit odd we don't mention tensors at all in this article. Mdwh (talk) 00:28, 5 April 2008 (UTC)

[edit] When a matrix is a vector?

The page currently says: A matrix where one of the dimensions equals one is often called a vector. This was supposed to be the same as the last edition (according to Oleg Alexander): "A matrix where only one of the dimensions is higher than one". However, both phrases does not say the same, because the first one only works for bidimensional matrixes, but what happens with matrixes that have three dimensions? The second sentence is therefore more correct. Example: an 3x2x2x1 matrix would be called 'vector' according to the first sentence, however it is not. —Preceding unsigned comment added by 88.6.21.192 (talk) 23:39, 23 March 2008 (UTC)

A matrix has only two dimensions. Certainly that is the meaning of matrix used in the whole article. A 3x2x2x1 "matrix" would usually be called a tensor (or an array in computer science). -- Jitse Niesen (talk) 23:57, 23 March 2008 (UTC)

[edit] Square matrix section

At the top of the article:

''For the square matrix section, see [[Matrix (mathematics)#Square matrices and related definitions|square matrix]]''. <!-- please do not remove this sentence. The page [[square matrix]] redirects here. -->

Contrary to the comment, I don't think this should stay. Square matrix already redirects straight to that section.

CRGreathouse (t | c) 13:05, 9 April 2008 (UTC)