Talk:Invertible matrix

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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)

The purpose is to say that singular (non-invertible) matrices are very very very rare. If you choose a matrix with random real entries (say, between 0 and 1), then the probability it is singular is literally zero. That is not to say that non-invertible matrices can't happen, just that they are infinitely unlikely. (if this seems like a contradiction, consider throwing a dart at a dartboard - what is the probability that the dart will hit a particular point?) 67.9.148.47 (talk) 11:50, 28 November 2008 (UTC)

I think this information is important to include, but that earlier in the article we can describe it in simpler terms, such as those used by 67.9.148.47 above. Dcoetzee 08:46, 29 November 2008 (UTC)

I think it is important to include that singularity is rare. Maybe we should add something small to make sure we're not overgeneralizing. Singularity over a euclidean field is rare. But isn't a square matrix over z2 almost always singular? — Preceding unsigned comment added by 132.235.46.80 (talk) 04:05, 9 November 2011 (UTC)

Prompted by the discussion here, I have moved it to later. It does not seem to be important enough to be in the lead of the article. Incidentally, the lead doesn't exactly comply with the WP:LEAD guideline anyway. It should probably be rewritten and the existing content relocated elsewhere. siℓℓy rabbit (talk) 13:53, 29 November 2008 (UTC)

Could someone add a simple example with real numbers? That was what I was looking for, just some actual example not involving variables. — Preceding unsigned comment added by 60.166.111.122 (talk) 14:19, 16 September 2015 (UTC)

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)

I agree that this article doesn't need an explicit formula for every single dimension, but I still would include a formula for the 3x3 case. Not the one above, which is admittedly long winded & inconvenient, just asking for making copying errors. But there is a much simpler one, only using cross product & triple product:
If an invertible matrix A consists of the column vectors $\mathbf {x_{0}} ,\;\mathbf {x_{1}} ,\;\mathbf {x_{2}}$, its inverse consists of the row vectors $\mathbf {x_{1}} \times \mathbf {x_{2}} ,\;\mathbf {x_{2}} \times \mathbf {x_{0}} ,\;\mathbf {x_{0}} \times \mathbf {x_{1}}$, multiplied by the inverse determinant of A, where the determinant just "happens" to be equal to the triple product of x0,x1 & x2: $\det(A)=\mathbf {x_{0}} \cdot (\mathbf {x_{1}} \times \mathbf {x_{2}} )$. That this matrix is a left inverse of A can be checked easily by using basic properties of the cross & triple products. And since left inverses & right inverses are identical for all groups, this is indeed the inverse of A.
If no one objects, I'm gonna include the formula. Catskineater (talk) 03:42, 6 February 2010 (UTC)

No one protested, so i added the formula. Catskineater (talk) 22:42, 21 February 2010 (UTC)

You guys should probably switch the matrix formula for the inversion of 3x3 matricies on the wiki page.

i.e

$\mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\\\end{bmatrix}}^{-1}={\frac {1}{Z}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,K\\\end{bmatrix}}$

where

$Z=a(ek-fh)+b(fg-dk)+c(dh-eg)$

which is the determinant of the matrix. If $Z$ is finite (non-zero), the matrix is invertible, with the elements of the above matrix on the right side given by

${\begin{matrix}A=(ek-fh)&D=(ch-bk)&G=(bf-ce)\\B=(fg-dk)&E=(ak-cg)&H=(cd-af)\\C=(dh-eg)&F=(bg-ah)&K=(ae-bd)\\\end{matrix}}$

I'm not 100% sure, but i'm pretty sure that that gives out the solution for the transpose of the inverse, not the inverse itself. A pretty easy and quick fix, but i'm pretty lazy and don't have the time to formulate it into wiki and make sure its all right. — Preceding unsigned comment added by Njc69 (talk • contribs) 16:35, 3 October 2010 (UTC)

I agree with the above. The formula on the main page caused me lots of problems since it is actually transpose of the inverse, but the above formula seems to work. — Preceding unsigned comment added by 65.60.221.79 (talk) 10:59, 8 March 2013 (UTC)

The formula on the page has been corrected more than two years ago, so you are probably misreading something.—Emil J. 12:25, 8 March 2013 (UTC)

I have only a moderate objection to the above cross/dot product representation: it's useless. If what we're looking for is a REPRESENTATION of the inverse of a matrix, then A^-1 is hard to beat. I'm writing this 6½ years after Catskineater's post, but I don't see how an alternative representation helps either clarify the topic or aid in computing the terms of the inverse matrix. I found the inclusion of the specific element by element expansion of the 3x3 matrix quite useful. I also strongly disagree with those who claim such a representation is prone to errors. That is, assuming copy and paste is something you are able to handle it is trivial and NOT "asking for making copy errors". I strongly doubt whether either the information about the Cayley-Hamilton decomposition or the information about the representation in terms of 3 column vectors is useful to 99.99% of the readers. I propose the 3x3 section be shortened by removing all the material after the Det(A)=aA+bB+cC AND I also propose abandoning the use of the elements A,B,...,I. They add very little in terms of conciseness; compare Det(A) = a(ei-fh)- b(di-fg)+c(dh-eg) with the above, there is very little space savings but the cost of including 9 more (extraneous) variables is significant in terms of clarity and simplicity. What purpose does it serve?71.30.36.108 (talk) 00:31, 9 August 2016 (UTC)

Matrix inverses in MIMO wireless communication

The statement that "It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information." is just plain wrong. The matrix H is not always square, and there are several better ways of decoding the transmitted signal, instead of inverting the channel matrix. I have never edited a Wikipedia page before, so I will figure that out before I make some correcting changes. — Preceding unsigned comment added by 89.160.119.209 (talk) 11:00, 13 December 2013 (UTC)

No mention of linear independence anywhere

A square matrix is singular if and only if it's determinant is zero.

A family of vectors are linearly independent if and only if the determinant of their matrix is zero.

I really think there should be a mention of linear independence here after the reference to singular matrices since singularity is equivalent to linear dependence, which also ties in to the discussion later about eigenvectors and such. There's already a page on linear independence so just a quick link would be nice.

SomeHandyGuy (talk) 00:33, 4 June 2014 (UTC)

Article Name: why not inverse matrix or matrix inverse ?

I think those phrases would appear more in text, i.e. someone reading and clicking wants to know what is an "inverse matrix", or how do I invert a matrix. Would it make sense to rename and reword this article? :-

"The inverse of a matrix 'A' is a matrix 'B' such that ... AB=I ; a matrix for which such an inverse exists is called 'invertible'..." Fmadd (talk) 08:38, 4 February 2017 (UTC)

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Formula not showing correctly

Somehow the formula for the inverse of a general 2x2 matrix is not showing correctly. The latex has a minus before the c but the output does not. I can't seem to fix it.— Preceding unsigned comment added by 92.194.124.84 (talk) 15:18, 14 January 2021 (UTC)

Thank you for your concern about the minus signs, but Wikipedia has a software bug. Editors have tried several times to kludge it to no avail. This has been reported multiple times now on WP:VPT. We'll just have to wait until the administrators fix it properly.—Anita5192 (talk) 16:57, 14 January 2021 (UTC)

Theorem, Explanations, and Applications

I will reorder the theorems to show audiences that they can be explained by each others first. In addition, the formulas are hard to most audiences for understanding, so I will add some explanations on these formulas. Also, Invertible Matrix can be used in many concrete ways in real life but the application part only talks about few of it. In sub-title least square solutions, the invertible matrix is always used for data analyzing for predicting future data, and people always used it to create model of the relationship between the variables and output. For example, to analyze the price of house, the variables should be area, layout, swimming pool, and place and the output should be price. So, I will add more content on application part.

"Nonsingular" listed at Redirects for discussion

The redirect Nonsingular has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 April 25 § Nonsingular until a consensus is reached. 1234qwer1234qwer4 23:04, 25 April 2023 (UTC)