Talk: Matrix (mathematics)
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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)
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)
| Contents |
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)
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)
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
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)
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)
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)
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)
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)
