Talk:Invertible matrix

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Mid Priority  Field: Algebra
One of the 500 most frequently viewed mathematics articles.

Contents

[edit] Sigh

Sorry to say so, but I think this page is too complicated.

Example from the introduction: "Over the field of real numbers, the set of singular n-by-n matrices, considered as a subset of R^{n \times n}, is a null set, i.e., has Lebesgue measure zero. (This is true because singular matrices can be thought of as the roots of the polynomial function given by the determinant.) "

Who cares? Sure, this may be an interesting aside, but it obfuscates more important things. In particular, since the "matrix inverse" page has been merged here, the extremely useful content that used to reside there should be clearer.

I have referred to Wikipedia's "matrix inverse" page innumerable times in the past to remind me how to do simple matrix inversion. (I forget these things!) Now when I look for that simple information I get overloaded with nonsense about the Lebesque measure of the set of singular matrices. Seriously?

Wikipedia used to be a good source for the basic explaination. If I needed more, I'd go to Mathworld- which was not very often, since I never understood what Wolfram was saying!

My request to the fantastic math heads writing this page: put the high-school stuff first, since that's what a lot people will want to see. Make it clear to people with minimum math background, and keep it simple when possible.

Thanks for the hard work. I'm just trying to help make the information useful for everyone.

Hawkeyek (talk) 05:51, 10 April 2008 (UTC)

[edit] Statement equivalence

The formulae given for inverting 2x2 and 3x3 matrices are not valid for matrices with non-commutative elements and maybe this should be made clear. On the other hand, there are many articles on matrices and to qualify every statement with words like 'where the matrix elements are real, complex, or multiply commutatively' would make the pages cumbersome for most readers. Perhaps a footnote?

I cancelled out these two lines beacause they are not equivalent with the invertibility of a matrix. Only if both are true and this is already given in the next line (..exactly one solution..).

  • The equation Ax = b has at most one solution for each b in Kn.
  • The equation Ax = b has at least one solution for each b in Kn.

I also removed

  • The linear transformation x |-> Ax from Kn to Kn is one-to-one.
  • The linear transformation x |-> Ax from Kn to Kn is onto.

beacuas these are not equivalent. And also not equivalent with the other statements

What? I can prove that the transformation is both onto and one-to-one for square matrices. These two statements are, in fact, equivalent iff A is a square matrix. The onto statement is equivalent to saying that Span{Col A}} = Kn, whereas the one-to-one statement implies that Nul A = {0}, both of which are given in the parts that are listed. If you want the full formal proof, you'll have to wait until I get my Linear Algebra textbook out. IMacWin95 01:36, 28 April 2007 (UTC)
Agreed. Oli Filth 11:56, 28 April 2007 (UTC)

[edit] What?

Are large parts of this article copy and pasted from answers.com, or did answers.com take large parts of this article? Can someone explain the statement: "As a rule of thumb, almost all matrices are invertible. Over the field of real numbers, this can be made precise as follows: the set of singular n-by-n matrices, considered as a subset of , is a null set, i.e., has Lebesgue measure zero. Intuitively, this means that if you pick a random square matrix over the reals, the probability that it will be singular is zero."

Does this mean the matrix

(0, 2)
(0, 0)

is invertible, even though it is not row reducible to the identity? Somehow I don't think so, but for a person not steeped in mathematical know-how, it is misleading and suggests that is indeed the case. Basically, this article messed me over because I used it as a study help. Someone who knows what they're talking about needs to re-write it, or else sue answers.com.

18.251.6.142 08:41, 10 March 2006 (UTC)

Answers.com copied this article under the GFDL, so all is fine. :)
That matrix is not invertible, and it is not row reducible either, so I don't see a problem. That text just says it is more likely that a given matrix is invertible that it is not, it does not mean all matrices are invertible.
I suggest you go over things which you don't know and read only the parts you understand. The article is written such that it gives some information both to people who know nothing abou this stuff, and to people who know a lot, it is not tailored specifically to you. If overall this article manages to answer some of your questions, I guess you should be happy with that. Oleg Alexandrov (talk) 17:41, 10 March 2006 (UTC)
Thank you Oleg. I am sorry if I seemed a bit upset, but other parts of this site have been very helpful in my coursework and this particular statement seemed a bit misleading and cost me a lot of time. 18.251.6.142 17:59, 10 March 2006 (UTC)

[edit] Invertible != regular ?

The article currently equates invertibility and regularity with the statement beginnging "In linear algebra, an n-by-n (square) matrix A is called invertible, non-singular, or regular if..." in the first line. However, this interpretation of "regular" conflicts with the definition "A stochastic matrix P is regular if some matrix power P^k contains only strictly positive entries" given on the Stochastic matrix page. I've not seen the term "regular matrix" before, but it seems that that page uses "regular" where I would use "ergodic", while this page uses it as a synonym for "invertible".

As far as I can see, one of the definitions must be wrong, as each can trivially be shown to exclude the other. In the unlikely event that both meanings are in common use, then it is an error on the Stochastic matrix page that "regular" is linked to this page. 152.78.191.84 12:08, 13 March 2006 (UTC)

I'm struggling with this seeming contradiction as well. I can't find any references to regular matrices in my (somewhat limited) library, but I found a few conflicting references online. Wolfram's site (http://mathworld.wolfram.com/RegularMatrix.html) just redirects me to the entry for "Nonsingular Matrix", which gives credence to the equivalence of invertability and regularity. I've found a few other definitions as well which don't seem quite as reputable. A book transcript from Springer-Verlag (http://www.vias.org/tmdatanaleng/hl_regularmatrix.html) gives the definition found on the Stochastic matrix page. Finally, Thinkquest (http://library.thinkquest.org/28509/English/Transformation.htm) gives the definition of a regular matrix as a matrix whose inverse is itself (A − 1 = A). This definition seems pretty off to me; wouldn't that imply that the only "regular" matrix is the identity or a permutation matrix?
I looked around for references on the Stochastic Matrix Theorem cited on the Stochastic matrix page but was unsuccessful as the page gives no references. Perhaps someone with more knowledge could point to a source for this theorem, which likely would clarify the confusion? Mateoee 17:54, 17 November 2006 (UTC)
I never heard of invertible matrices being called regular. I will remove that from the article. Oleg Alexandrov (talk) 02:45, 18 November 2006 (UTC)

[edit] Rephrase a sentence?

I find the sentence "The equation Ax = 0 has infinitely many the trivial solutions x = 0 (i.e. Null A = 0)." very confusing. Why not rephrase it to "The equation Ax = 0 has only the trivial solution x = 0."? If the former sentence is more correct I apologize for my lack of knowledge in this subject. :/ Karih 18:18, 3 December 2006 (UTC)

You're absolutely right, it is very confusing to put it mildly. I fixed it; thanks for bringing this to our attention. -- Jitse Niesen (talk) 02:10, 4 December 2006 (UTC)

[edit] Inversion of 3 x 3 matrices

please!

I understand why you would not wish to post a general form for the inversion of a 3x3 matrix, but maybe a step by step with a simple example? Nightwindzero 05:52, 22 February 2007

There are links to two different methods for solving systems that involve inverse matrices, as well as a description of the general analytic method for obtaining the nxn inverse. It would be completely unnecessary to show an example of 3x3 in this article, IMO. The only reason that the 2x2 is shown is because it's trivially simple. Oli Filth 10:54, 22 February 2007 (UTC)

From http://www.dr-lex.34sp.com/random/matrix_inv.html:

A^{-1} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}^{-1} = \frac{1}{a(ie-hf)-d(ib-hc)+g(fb-ec)} \begin{bmatrix} ie-hf & hc-ib & fb-ec \\ gf-id & ia-gc & dc-fa \\ hd-ge & gb-ha & ea-db \\ \end{bmatrix}

Hopefully I copied it over rightly. Looking at the letters like this makes the pattern of 2x2 matrices excluding the row and column of the element in question, turned sideways, seem much more intuitive, although it sure sounds complicated when I say it out like that. 72.224.200.135 02:44, 12 June 2007 (UTC)

[edit] Inversion of 4 x 4 matrices

please!

Although it is possible to derive equations for the inversion of 3x3 and 4x4 matrices like the one for the 2x2 matrix, they will be huge (and therefore not really suitable for the article). The generalised analytic form is already given (i.e. in terms of determinant and co-factors), and furthermore, inversion may be achieved more practically using an algorithm such as Gaussian elimination. Oli Filth 09:01, 19 January 2007 (UTC)

[edit] Note

I think making a statement like "As a rule of thumb, almost all matrices are invertible" is vague and not accurate. There may be more invertible matrices than not, but a statement like that will certainly confuse many readers, especially those that are new to the subject. 69.107.60.124 18:31, 28 January 2007 (UTC)

I agree. I will remove that. Oleg Alexandrov (talk) 23:15, 28 January 2007 (UTC)
Actually, after reading the text, I disagree. That statement is definitely not precise, but it is made precise in the next sentence, and that rather vague statement is used to motivate the numerical issues below.
I don't much like the current intro, but it has its good points and I can't think of anything better to replace it with. Oleg Alexandrov (talk) 23:19, 28 January 2007 (UTC)
I think that "As a rule of thumb" is misleading. (At least one of my students was slightly confused by it.) I deleted that and rephrased/reordered parts of the paragraph. Hopefully, the new version is less confusing. Fgdorais 21:02, 17 September 2007 (UTC)

I would like to add that "almost all square matrices are invertible" can also be interpreted in the sense of category, i.e., the set of invertible matrices is open dense. This is more intimately connected with perturbation of coefficients and numerical considerations. Perhaps this material should be removed from the introduction and have its own paragraph where these two (and pehaps other) interpretations can be discussed. I may decide to write such a paragraph when I'm less busy, but I would be very happy if someone else were to volunteer. Fgdorais 14:55, 23 September 2007 (UTC)

[edit] Account for other systems than R

in "Inversion of 2 x 2 matrices", 1/(ad-bc) is used. Should it be (ad-bc)^-1 to account for other systems such as rings ? I'm in no way a mathematician so i ask someone more knowledgeable to consider. Dubonbacon 18:17, 23 February 2007 (UTC)

I'd guess that people sufficiently advanced to know about such stuff will readily convert between these notations. I think that one would probably need to assume that the entries come from a field instead of a ring. But I'm in no way a pure mathematician so all my matrices have numbers in them. -- Jitse Niesen (talk) 12:34, 26 February 2007 (UTC)

[edit] Gaussian elimination example

I've reverted the addition of an example of Gaussian elimination, because that is already covered in the Gaussian elimination article. That article would be the appropriate place to add an example. This article is concerned with the mathematics of inverse matrices, not the numerical intricacies of how to obtain them. (Otherwise, for parity, we'd need step-by-step numerical examples of Newton's method and LU decomposition as well, which would hideously bloat the article.)

Adding an example that has no explanation of the steps involved is certainly not helpful! Oli Filth 23:44, 3 May 2007 (UTC)

[edit] "regular" revisited

Hello

As I could find in a former discussion (2006), the term "regular" has been removed from the introduction because it seems not equal to "inversible". But still "Regular matrix", as it is referred to in "Irregular matrix", e.g., redirects to this article. First, I was quite confused that the opposite of a matrix with "a different number of elements in each row" should be an inversible matrix. Second, the more confused I was when I could not find the term "regular" anywhere in the whole article where I was redirected from "Regular matrix"...

--chiccodoro —Preceding unsigned comment added by 131.152.34.104 (talk) 09:29, 26 May 2008 (UTC)