Talk:Matrix theory

Definition?

Would it be more correct to say that the value of a matrix can be represented as a rectangular array of numbers rather that it is a rectangular array? Mathematics extracts an abstract structure from matrices and studies these properties, so we can have a matrix without having numbers written down, right? (I'm probably getting pedantic but it's more than just a bunch of numbers). RJFJR 22:04, Dec 25, 2004 (UTC)

← I couldn't agree more. =)

--Cawas 04:34, 16 November 2005 (UTC)

Algebra of matrices

I added some material on the algebra of matrices. Then I tried to use that material to show how matrices are used. In other words I want to show why matrices are interesting under the theory rather than just what they are (which is at matrix (mathematics). RJFJR 22:04, Dec 25, 2004 (UTC)

Subscripts inside math tag?

X = <math>\begin{bmatrix}x₁\\ x₂\\ x₃ \end{bmatrix}</math> results in X = ${\begin{bmatrix}x₁\\x₂\\x₃\end{bmatrix}}$

Is there a way to use subscripts in a <math> tag? RJFJR 22:10, Dec 25, 2004 (UTC)

By the way,

to answer your question:

Is there a way to use subscripts in a <math> tag? RJFJR 22:10, Dec 25, 2004 (UTC)

The answer is yes! See the source of this:

$A={\begin{bmatrix}a_{0}&a_{-1}&a_{-2}&\ldots &\ldots &a_{-n+1}\\a_{1}&a_{0}&a_{-1}&\ddots &&\vdots \\a_{2}&a_{1}&\ddots &\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &\ddots &a_{-1}&a_{-2}\\\vdots &&\ddots &a_{1}&a_{0}&a_{-1}\\a_{m-1}&\ldots &\ldots &a_{2}&a_{1}&a_{0}\end{bmatrix}}$

(stolen from Toeplitz matrix. Oleg Alexandrov 04:44, 7 Jan 2005 (UTC)

Matrix (mathematics) vs. matrix theory

See Talk:Matrix (mathematics)#Matrix (mathematics) vs. matrix theory for some discussion. Oleg Alexandrov 23:09, 31 Dec 2004 (UTC)

Algebra of Matrices and application to simultaneous equations

The material on Algebra of Matrices was intended to demonstrate the sort of thing that can be developed for matrices without specific attention to what a matrices 'is', it applies to the abstract concepts that algebras apply to. The material on matrices for solving simultaneous equations was intended to demonstrate the motivation for matrices and matrix theory; it was intentionally a simple example. RJFJR 04:10, Jan 7, 2005 (UTC)

Before I comment on why I deleted that contents, let me remark that one does not write "Algebra of Matrices", but rather "Algebra of matrices" (downcase "m"). This is important; unfortunately I see this mistake endlessly on Wikipedia.

Now, to my remarks. You see, the word "algebra" has many meanings. In math, an algebra is a ring which is a module over another ring (see algebra over a field). So, the square matrices with real entries are an algebra over the reals, because you can add and multiply matrices, and you can multiply them by real numbers. The set of all matrices is not an algebra, because not any two matrices can be multiplied. So the title was kind of misleading.

You wrote in Talk:Matrix (mathematics) that

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 am not sure that is true. I think "Matrix theory" is about abstract and general properties of matrices. And solving a linear system is not that. But I could be wrong.

Anyway, I felt such an elementary thing as solving a linear system does not fit well in the same article as the much more abstract topics on top of this article (topics which are put in the section "Introduction" even if that is not an introduction, it is just a list of facts.). But this is my opinion, you might think otherwise.

I honestly don't know how to fix the Matrix theory article. If you want, you can put the linear system back. But either with the system, or without it, I think this article is fundamentally broken. It does not have coherence, or purpose. I dont't know how to fix it. 04:37, 7 Jan 2005 (UTC)

I moved the part about answering subscripts inside a math tag to the above section. RJFJR 18:33, Jan 7, 2005 (UTC)

The part about solving a linear system was intended as motivation (it's practical) and as an example to give me an excuse to put in some about how (some) matrices have inverses and you can multiply by them and cancel them and such. It may not have been a good way to show that you can manipulat the representation for matrices. (I almost added LUP decomposition, without how to find the decomposition, as an further example of matrix manipulation but (even) I felt that was too far off.) RJFJR 18:33, Jan 7, 2005 (UTC)

Should we merge Matrix theory into Matrix (mathematics)?

There isn't much left of the Matrix theory article. What is left could probably be moved into matrix (mathematics).

I suggest we do that and make matrix theory a redirect to matrix (mathematics). It saves us the trouble of trying to figure out what goes in each article. The only danger is the combined article getting too big. RJFJR 18:33, Jan 7, 2005 (UTC)

Today I got a message which I paste below. Oleg Alexandrov 19:07, 7 Jan 2005 (UTC)

Matrix vs matrix theory

Hi, I have noticed your discussion on Talk:matrix (mathematics). I am having similar problems with graph (mathematics) and graph theory. Perhaps you want to have a look at Wikipedia_talk:WikiProject_Mathematics#Graph_.28mathematics.29_vs_Graph_theory where I started a discussion on this subject. And welcome by the way :) MathMartin 12:40, 7 Jan 2005 (UTC)

I would suggest you put the link above on your watchlist, and you could follow that link and maybe we can start a discussion. Oleg Alexandrov 19:07, 7 Jan 2005 (UTC)

I think we should not merge the articles. Rather, some things from Matrix (mathematics) which are more complicated should come here. What do you think could be transfered here? Oleg Alexandrov 12:15, 9 Jan 2005 (UTC)

I think matrix theory should be merged into matrix and matrix should be renamed into matrix theory. Then perhaps certain parts of this new article should be put into subarticles. See my latest posting at Wikipedia_talk:WikiProject_Mathematics#Graph_.28mathematics.29_vs_Graph_theory. MathMartin 15:19, 9 Jan 2005 (UTC)

If you want to divide the material into two articles, a basic one and a more sophisticated one, the pages should be named accordingly (like elementary matrix theory <-> matrix theory and not matrix theory <-> matrix). But I think this is a bad idea and instead the article should be structured in such a way as to present a basic, accessible definition at the top and more complex definitions and topics later. MathMartin 16:02, 9 Jan 2005 (UTC)

What you say makes sense (both options you suggest). I agree. Would you be willing to do the reorganization? Oleg Alexandrov 01:13, 10 Jan 2005 (UTC)

Sorry, somehow I missed your reponse. I probably won't do it in the near future. At the moment I am more interested in graph theory topics. MathMartin 12:08, 1 Feb 2005 (UTC)

I thought moving things myself. But somehow I think things are fine the way they are. Matrix theory is a summary of more advanced things, while Matrix has more elementary properties. So I would be inclided not to do any big changes on these pages. If you have other suggestions, I could implement them. Oleg Alexandrov | talk 16:24, 1 Feb 2005 (UTC)

It's been four years, so I don't know if any of you are reading this, but at least right now, Matrix contains not only the elementary definition and properties of matrices, but also a lot of more advanced material, including a list of many applications. And I see nothing wrong with this. But it does mean that most of the material on the Matrix theory page is now redundant. Also, different people use the term "matrix theory" to mean different things: for some, it is just the basic properties of matrices; for others, it is synonymous with linear algebra. It is by no means a consensus or even a majority that interprets "matrix theory" to mean "the advanced aspects of matrices". Given all this, there is a strong case for merging the pages. As for whether to call it "Matrix" or "Matrix theory", I don't have a strong opinion. If I had to choose, I'd say that "Matrix" sounds better, more fundamental, as a title. More people are likely to search for the "Matrix" page than for the "Matrix theory" page. Of course, whichever one is removed should redirect to the other. FactSpewer (talk) 23:17, 18 April 2009 (UTC)

What a matrix may be

It seems there is more than one person saying a matrix is more than an array of numbers. I find this definition very satisfactory though. After you are very comfortable with this definition, you of course can move on to higher views of a matrix.

The second paragraph in matrix theory states that

A matrix can be identified with a linear transformation between two vector spaces. Therefore matrix theory is usually considered as a branch of linear algebra. The square matrices play a special role, because the nxn matrices for fixed n have many closure properties.

People who like to see in a matrix more than just a humble table of numbers, could make this paragraph into a stand alone section, and explain here the higher view of matrices. I think it is always better to start with the simplest definition possible (as for a matrix to be a 2D array) and then make things more complex later. Any comments? Oleg Alexandrov 17:41, 19 Jan 2005 (UTC)

I do agree it's usually better start with the simplest definition possible, but to be able to properly do that you have to be careful with choosing each word, each letter.

So, sometimes, when you are not sure about word definitions (or communication protocol standards) it might be better start with the most far from simple definition because that way you don't have to be too careful with each word. Since you are not talking about a basic thing you can forget a little the fact that on a basic statement each single letter could lead to miss understanding, or ambiguous thoughts, if it's not carefully choosed.

Anyway, bottom line, for the article, it's good to start with simple but it's also good to get some steps and also include information on the most far we could go from the basic.

While I believe matrix (mathematics) won't go too far, I think the matrix theory can get really surprising.

--Cawas 04:31, 16 November 2005 (UTC)

matrix - main article

I'll be a little invasive in here and change one "little" aspect in the main article.

It's also my answer for the first question in this discussion, and I quote:

Would it be more correct to say that the value of a matrix can be represented as a rectangular array of numbers rather that it is a rectangular array? Mathematics extracts an abstract structure from matrices and studies these properties, so we can have a matrix without having numbers written down, right? (I'm probably getting pedantic but it's more than just a bunch of numbers). RJFJR 22:04, Dec 25, 2004 (UTC)

Yes, I believe it is more correct.

Matrix theory to me talks about matrices using logic rather than mathematics. After all, it is being compared to matrix (mathematics)

Logic can be expressed in mathematics and many other fields. In this context, an encyclopedia, I believe we are talking about theories using english. So, that should be the field used to explain matrix theory.

So this also would answer Oleg's question about the very interesting question on merging or not both Articles. I agree it's not even good not to merge, as it would be better expand the main article into matrix theory and use it to link others. Also we could change the matrix disambiguation page to make it clear what's the main article.

So I'll basically just change the main article to update this idea.

The Movie

I got in here, Matrix Theory Discussion Page, through the movie. After reading and analyzing thousands of thoughts about all the theory evolving matrices I realise it have being really connecting me in the way I understand the existence of life and any other thing.

Don't get away thinking this is irrelevant. I believe, through deep logic, it is more relevant that it may look like at first sight.

I believe the matrix theory could actually explain the whole Matrix in the movie, and in so many other histories that talks about the same idea. A whole universe generated by machines? Nope. Just about other universes. The little I do know about matrix (mathematics), and its respective basic operations, actually tells me about a mathematical way to do operations between universes.

I'm just wondering if anyone else got somewhere around this point of matrix theory. I might be going way too far, but I do feel it is the right direction of expanding the theory.

--Cawas 04:23, 16 November 2005 (UTC)

I don't think the matrix theory has anything to with The Matrix. There, I think they use the word "Matrix" to mean "womb" or something like that (as translated from Latin).

And no, a mathematical matrix does not do operations between universes. It is just a linear map between linear spaces. The universe is much more complicated than that, and serious map between universes I think would not be so elementary as a matrix vector multiplication. Oleg Alexandrov (talk) 04:47, 16 November 2005 (UTC)

Thanks for your reply, but I think you haven't really stop to get a deeper thought on this subject.

I do understand perfectly how simple the matrix (mathematics) is if compared to an universe. I'm talking about thinking beyond that, as a basic line on how things could happen. A matrix isn't necessarily bi-dimensional. We just think about it that way, in mathematics, to be able to use "very simple" matrices operations with them. They are complex, but way more simple than the complexity a universe is.

As about the movie and a womb, yeah, you can get that impression from the movie. But in reality that matrix is a parallel universe. You probably haven't watched it too much, the matrix in the movie is our world as in 1999 re-created by the machines to control humans to be able to pacifically extract energy from our body. It is, of course, not physically possible, but then again, if all physics we know is what they want us to know, that could just be another control element.

If the Matrix we study in mathematics has anything to do with universes, and I can almost be sure it does, then it is more related to the movie than the impression we have from the title and its association with womb, virtual reality or whichever minor things we might have at a first impression.

Allow me add some definitions to the main article and let's see how people react to that... I haven't done this before because I got stuck with a double virus in my life, and my firefox crashed when I was about to submit it. Sorry if this comments is too irrelevant. :P

Anyway, once again, thanks for your prompt answer and caring, Oleg! :)

--Cawas 06:54, 18 November 2005 (UTC)

Please don't add anything to the article yet, I am not sure you know what you are saying.

Making a matrix higher dimensional does not change the fact that it is still a very very simple mapping. Too simple to even describe an atom, not to talk about the whole universe.

Just yesterday I attended a talk about supersymmetry. The amount of mathematics involved in describing the universe is staggering. Your belief that a matrix is helpful in describing the universe is like a kid's dream that using his toys he can build a skyscraper. Oleg Alexandrov (talk) 16:32, 18 November 2005 (UTC)

True. About the universe being described as a Matrix I totally agree with you. Even because I am just a kid. So it's probably as you say. I was just trying to figure out what other people would think about it. Eventhough I'm quite expecting this subject to die here, I would still welcome more informed opinions.

And about changing the article, it's already done. It's not really directly related to my idea of universe. It's just a non-mathematical way to describe a Matrix since for mathematical studies of it we already have matrix (mathematics) ;).

But that should be "discussed" on the other topic.

--Cawas 16:51, 18 November 2005 (UTC)

Split into Matrix Theory (math) and Matrix Theory (film)?

Lilar recently edited parts of this article transforming it into an amusing and informative parody of its former self.

Should the pre-Lilar changes be kept as Matrix Theory (math) and the parody as Matrix Theory (film)?

—Xojo (talk) 05:55, 18 July 2008 (UTC)

• Don't forget Matrix theory (physics). 78.102.107.177 (talk) 17:38, 26 November 2010 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Matrix theory/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

* Needs history, motivation, much more on applications, oeverveiw fo theroems linked to in "See also"... Tompw 14:02, 5 October 2006 (UTC) With the current content (essentially a subset of Matrix(mathematics)) the article is more of a stub than a "start". In addition, at least the current content fails to clarify why an article separate from Matrix(mathematics) and linear algebra is needed for "Matrix theory". If the article contributors disagree and have in mind High/Top-importance content which is not naturally treated under the aforesaid articles (as well in individual articles on topics such as Cayley-Hamilton theorem), please do revert my change. However, if no such topics emerge, I would suggest merging this with Matrix. Stca74 16:37, 14 June 2007 (UTC)

Last edited at 16:37, 14 June 2007 (UTC). Substituted at 21:51, 26 June 2016 (UTC)