Talk:Invertible matrix

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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

Contents 1 What? 2 Invertible != regular ? 3 Rephrase a sentence? 4 Inversion of 3 x 3 matrices 5 Inversion of 4 x 4 matrices 6 Note 7 Account for other systems than R

[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)

[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)

[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)