Talk:Determinant/Archive 1

Archive 1

Archive 2

Archive 3

Formula

It would be useful to add this formula

$|A+BCD|=|A||C||C^{-1}+DA^{-1}B|$

which I always think of as the equivalent of the Sherman-Morrison for determinants, i.e. it allows you to update a determinant without recomputation. However, I'm not sure of it's origins (I just have it written on a scrap of paper) or how general it is, so I'd rather someone with more knowledge put it up. --Lawrennd 14:03, 30 September 2005 (UTC)

See Matrix determinant lemma. --Lionelbrits 03:29, 4 October 2007 (UTC)

Requests

"The interpretation is that this gives the area of the parallelogram with vertices at (0,0), (a,c), (b,d), and (a + b, c + d), with a sign factor (which is −1 if A as a transformation matrix flips the unit square over)."

   Could someone draw up an example please? - Cyberman

You could draw it, upload it and add it... --Carbonrodney

Just added that picture. It's done:) Feel Free to edit mercilessly, perhaps with an animated gif. Rpchase 21:50, 7 November 2006 (UTC)

Maybe someone could add something about the computation costs of finding a determinant. I've heard of algorithms that are big O of n^d, with d<3 (d=3 for Gaussian Elimination) and specifically I've heard of a rumour that someone proved you can get d asymptotically as close to 2 as you want. Anyone know if this true? -Stephen

It should be possible to code it linear time. I will write one up after exams, maybe. --Carbonrodney

Somehow this article seems a little disorganized; maybe somebody has some idea to make it more structured. I think determinants are connected to many notions in linear algebra (invertibility, # of solutions etc), and to notions in other fields of mathematics (i.e. wronskian). Maybe somebody who is expert can add some more of these properties, or appropriate links. Also it seems there is a lot of determinant magic out there. Anton (28.02.07)

Misc

I find the entire page too in-depth. This is not necessarily a bad thing for an encyclopedia, but we should consider users who are not so math savvy, or users with just intermediate math who just came in to find out how find the determinant of a 3x3 matrix.

Finding a 3x3 determinant, non-math savvy description:

1. multiply the numbers on the diagonals that go left to right (imagine two of the diagonal wraps around the matrix) and sum the products.
   S1 = (a₁ · b₂ · c₃) + (a₂ · b₃ · c₁) + (a₃ · b₁ · c₂)
   
2. multiply the numbers on the diagonals that go right to left and sum the products.
   S2 = (a₃ · b₂ · c₁) + (a₂ · b₁ · c₃) + (a₁ · b₃ · c₂)
   
3. the determinant is the first sum subtract the second one.
   det[]=S1-S2
   

Also a picture or two wouldn't hurt. Something like http://mathworld.wolfram.com/Determinant.html is a lot more pleasant looking than pure text.

Adding a easy to understand section like my description above would expand the use of this Wikipedia page to being a math reference instead of just research material.

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No, no, no. Please never dumb our topics down like this. It's better to force someone up to your level than bring yourself down to theirs.

Anyhow, the explanation above of how to find the determinant of a 3x3 matrix is in itself information of very little use or value. It does nothing to explain what determinants are -which should be a crucial requirement of this topic. Also, what do you do if you wnat the determinant of a higher order matrix? A brilliant generalised explanation of how to calculate determinants can be found here...

http://people.richland.edu/james/lecture/m116/matrices/determinant.html —Preceding unsigned comment added by 90.203.33.5 (talk) 02:04, 29 April 2008 (UTC)

My only comlaint is: In the section headed 'Determinants of a 2x2 matrix', the penultimate sentence, which begins "With the more common matrix-vector product...." seems WRONG! Can someone correct this please.

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The last phrase on this page seems pretty suspect. What exactly is a "Linear Algebraist"? A specialist in Mult-Linear algebra?

On the other hand I have never met an algebraist who "preferred" the Leibnitz formula. I suppose it might be useful to compute in certain situations but I can't imagine one claiming that one sshould just forget everything else and remember that.

Somebody (myself, if I'll win the laziness) should add something about the formal definition of determinant (an alternating function of the rows or columns etc. ...), of which its unicity and how to compute it are consequences. --Goochelaar

...and add to that the foundation of the definition, which is something to do with multilinear functions.
Also worth mentioning that historically, the concept of determinant came before the matrix.

That would certainly be very interesting. What is the history of the concept? --AxelBoldt

I'll see what I can dig up, but briefly: a determinant was originally a property of a system of equations. When the idea of putting co-efficients into a grid came up, the term "matrix" was coined to mean "mother of the determinant", as in womb.
The determinant function is defined in terms of vector spaces. It is the only function f: F^n x F^n .... x F^n -> F that is multilinear & alternating such that f( standard basis ) = 1.
Obviously, the above needs a major amount of fleshing out....

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I rewrite the page in a format similar to trace of a matrix. Wshun

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Text moved over from Talk:Determinant mathematics

Perhaps mention of the Scalar Triple Product, a.k.a. the Box Product, is fitting in the paragraph about the volume of the parallelopiped. If only to introduce the nomenclature.

I'm not familiar with that. Is it just the determinant of three 3-vectors? --AxelBoldt

Essentially, yes. According to Advanced Engineering Mathematics by Erwin Kreysig: "The scalar triple product or mixed triple product of three vectors

  a = [a1, a2, a3],   b = [b1, b2, b3], c = [c1, c2, c3]

is denoted by (a b c) and is defined by

(a b c) = a · (b × c)."

Since the cross product can be defined as a determinant where the first row is comprised of unit vectors, it is easy to prove that the scalar triple product is the determinant of a matrix where each row is a vector. Take its absolute value, and you get a volume. Another use of the product, besides computing volumes, is to show that three 3-d vectors are linearly independent ((a b c) ≠ 0 => a, b, c are linearly independent). From what I understand, it's a dying notation because it can be described in terms of the dot and cross products, but it still has a couple of uses.

Perhaps just include mention of it on this page, and define it on a vector calc page.

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Hmmm - talk about determinants with vector entries - that really ducks what's going on, no? Which is a 2-vector (wedge of vectors) being paired with a vector. Charles Matthews

I moved this out of the page.

Here is a 2-by-3 matrix (used when taking the cross product of two vectors)

$B={\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}}$

which has the determinant (in vector form)

$\det(B)=[bf-ce,cd-af,ae-bd]$.

This is a bit off-topic, and confusing on a page about square matrices. It really belongs with (perhaps) cross product, or introductory exterior algebra.

Charles Matthews 18:45, 21 May 2004 (UTC)

I always knew that procedure having the three basis vectors in the first column, for a, b ∈ R³ ie

$B={\begin{pmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\\end{pmatrix}}$

which keeps the matrix square, and keeps the notation consistent too Dysprosia 00:31, 22 May 2004 (UTC)

But what does a determinant with vectors in it mean? This is a good mnemonic, though. Charles Matthews 08:21, 22 May 2004 (UTC)

I'm not sure that it has any other special (tensor-ish?) meaning, other than if one writes that determinant in terms of the Levi-Civita symbol having those basis vectors there help organize the components. But yes, it is a good mnemonic :) Dysprosia 00:15, 23 May 2004 (UTC)

I think this actually belongs at minor (linear algebra), as a concrete example to balance the general stuff. And this article should link there, in relation to taking determinants when the matrix is not square.

Charles Matthews 08:34, 23 May 2004 (UTC)

As a non-mathematician I hesitate to edit the page myself. I might suggest something that is all to frequently missing from discussions of mathematical concepts. That is an intuitive interpretation of what the concept means. Here is my suggestion.

First, the determinant of a scalar is simply the number itself. Next consider a two by two matrix with the off diagonal of zero. In this case we can consider the matrix to define a pair of perpendicular vectors, and the determinant is easily seen to be the area of the rectangle the vectors describe. When the off diagonal elements of the matrix are non-zero the determinant still defines the described area, but now it is the area of a parallelogram. Similarly, the determinant of a three by three matrix is the volume of the described parallelepiped, and a four by four or higher matrix the volume of the described "hyperparallelepiped".

The point of all this is to allow the lay (or nearly lay) user to gain an intuitive understanding that will allow them to interpret the equations within this intuitive framework. A statement of this sort would go a LONG way towards increasing mathematical literacy. —Preceding unsigned comment added by 132.198.177.113 (talk) 14:11, 5 October 2007 (UTC)

Alternate definition?

I think we could add the fact that the determinant is defined as it is because it is the only function

$F:\{n\times n\;matrices\}\longrightarrow \mathbb {K}$

with the properties:

• it is linear w.r.t. columns;
• whenever any two columns are exchanged, it changes its sign;
• F(Id) = 1.

If nobody disagrees, I will add this in a couple of days. Cthulhu.mythos 15:26, 29 May 2006 (UTC)

I think that is a fine idea. This "definition" requires a theorem (such a map is unique) before it is well-defined, but it is of course much more explicit about the useful properties the determinant should have. In fact, your description is nothing more than an expression of the "best" definition in terms of columns: the determinant is the induced map of some linear transformation on the top exterior power. Exterior powers are defined in terms of a universal property of alternating maps, and the definition you cite is none other than that. Anyway, it's a good idea. Don't wait a few days, go ahead and do it now. -lethe ^{talk +} 16:53, 29 May 2006 (UTC)

Done. As soon as I manage to reconstruct the proof, I will add it too. Cthulhu.mythos 16:25, 30 May 2006 (UTC)

Including a proof here would make the page even more overlong. I put it in Leibniz formula (determinant). Cthulhu.mythos 08:55, 31 May 2006 (UTC)

Thanks, the addition looks good. I don't think a proof is too important. What we need now is an explanation and example of how to calculate a determinant by row reduction, a calculation based on the alternating-ness. It's shameful that there's only a brief passing mention of this algorithm, since that's how it's actually done in practice. Only a dummy uses expansion by minors. -lethe ^{talk +} 10:57, 31 May 2006 (UTC)

Here is a very interesting alternate definition of the determinant: —Preceding unsigned comment added by 79.203.249.42 (talk) 14:24, 2 June 2010 (UTC)

Clarification

$\det(\exp(A))=\exp(\operatorname {tr} (A)).$

Doesn't this only hold when A is diagonalizable? No such limitation is mentioned. Can anyone prove this? (132.163.135.113 00:31, 12 December 2006 (UTC))

No, it always holds. This is exercise 6.2.4 in Horn and Johnson. Topics in Matrix Analysis; I'm sure it's also in many other books. The proof is via the Jordan normal form, I'd guess. Alternatively, take a look at the proof at PlanetMath. -- Jitse Niesen (talk) 06:49, 12 December 2006 (UTC)

Mistake in image

I think there is a mistake in the current version of Determinant.png. Please have a look at Image talk:Determinant.png. --Abdull 16:50, 1 January 2007 (UTC)

The same point was made by a remark left by an IP editor, who said:

"The drawing is wrong, exchange a and d for b and c. I mean, it is right, but if you don't exchange the letters, the area is negative, which is not wrong, but confusing."

-- Jitse Niesen (talk) 01:06, 23 March 2007 (UTC)

Please, correct this error... It shouldn't be too hard to exchange (a,b) and (c,d) in the figure. —Preceding unsigned comment added by 66.11.173.16 (talk) 21:17, 13 April 2008 (UTC)

It would be nice to see some actual numbers in image and matrix. As I have yet to figure out what numbers to use. 166.102.59.254 (talk) 19:05, 2 May 2008 (UTC)

I find it confusing that the matrix A has column vectors (a,c) and (b,d) while in the chart the vectors (a,b) and (c,d) are used. If you use the vectors (a,c) and (b,d) in the chart, the conclusion that the surface area of the parallelogram is the determinant of the matrix A still holds. I find that approach much more insightful than the current chart.

Using the site in French

The site in French is soo beautiful!! We could put some of its figures and the geometric interpretation of determinant explaining how the properties for determinants relates to properties of area and volume for 3x3 matrices. Unfortunately, I don't understand French. One approach would be to more or less translate those parts or to construct those parts from scratch. Any Ideas? —The preceding unsigned comment was added by Ricardo sandoval (talk • contribs) 08:04, 11 April 2007 (UTC).

Sarrus Scheme

Hi, I added a line in "general definition and computation" where it discusses how to calculate the determinant of a 3x3. I mentioned the Sarrus scheme. I'm really don t know how to edit wiki properly, but i think i would be good to have a drawing/scheme of sarrus law. Can anyone do it? —The preceding unsigned comment was added by 195.23.217.69 (talk) 10:59, 4 May 2007 (UTC).

Diagram: 3D, singular

There is one diagram showing a 2-D parallelogram. There should be one in 3-D and one showing a singular matrix, illustrating why that transformation can't be invertible.—Ben FrantzDale 17:10, 7 May 2007 (UTC)

I have added the 3d picture. Rocchini 09:54, 21 September 2007 (UTC)

Vertical bar notation

additionally, the absolute value of a matrix is, in general, not defined

I have found this not to be the case: the absolute value of a complex-valued matrix M is, generally, defined as the square root of M*M or of MM* (usually in the context of Polar decomposition). But I have not spent enough time with the article to be comfortable changing it. 128.135.100.101 04:47, 29 May 2007 (UTC)

I thought that "absolute value of the matrix A" means the entry-wise absolute value of the matrix (the matrix whose entries are the absolute values of the entries of A). I believe you that it's used in your meaning. However, in both meanings, the absolute value is defined, so I don't know what the sentence in the article is supposed to mean. Perhaps the author was thinking about matrices over something else than real or complex numbers? But then the matrix norm is also not defined.

Anyway, I removed the sentence. -- Jitse Niesen (talk) 18:44, 29 May 2007 (UTC)

French Featured Article

I could try using the French article to improve the English one. Have you any input? Besselfunctions 00:37, 5 July 2007 (UTC)

You could start with merging the history section. —Cronholm¹⁴⁴ 11:40, 21 September 2007 (UTC)

Discrepancy between image and written material

This may be irrelevant, but the image to the right presents the information differently than the written material beside it. The image shows six columns being used to calculate the determinate, however only five are used in the written material. I know that they come out to the same answer, however should it not be presented in the same way for simplicity's sake and to make it less confusing for people who are not familiar with the subject?

Here is the part of the article i am referring to:

The determinant of a 3x3 matrix can be calculated by its diagonals.

which can be remembered as the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements when the copies of the first two columns of the matrix are written beside it as below:

${\begin{matrix}\color {blue}a&\color {blue}b&\color {blue}c&a&b\\d&\color {blue}e&\color {blue}f&\color {blue}d&e\\g&h&\color {blue}i&\color {blue}g&\color {blue}h\end{matrix}}\quad -\quad {\begin{matrix}a&b&\color {red}c&\color {red}a&\color {red}b\\d&\color {red}e&\color {red}f&\color {red}d&e\\\color {red}g&\color {red}h&\color {red}i&g&h\end{matrix}}$

Sirtrebuchet 04:54, 8 November 2007 (UTC)

Alternate characterizations

The article already mentions various properties of the determinant, but it's significant that some of these can be used as alternate definitions/characterizations. I wonder if it would be useful to have a separate section that brought together those characterizations (e.g. volume of an n-box, volume distortion factor, product of eigenvalues, constant term in characteristic polynomial, product of pivots, unique function respecting row operations in a certain way) and emphasized that various authors use them as their definition, as opposed to the multilinear property used as the definition here. Comments? -- Spireguy (talk) 23:21, 6 March 2008 (UTC)

2x2 matrix

There is a mixup in the 2x2 matrix. They have the direction of the solution wrong. According to me the formula should be: [ac]= ad-bc [bd]

instead of

[ab]= ad-bc [cd]

The solution of their matrix should be

[ab] = ad-cb [cd]

I'm pretty confident about this direction, because when the method is expanded in a 3x3 matrix one works from left top to right bottom, then from left bottom to right top.

Jan De Neys 212.190.66.254 (talk) 09:56, 18 July 2008 (UTC)

We're working over commutative rings here, so it makes no difference. Algebraist 01:05, 22 August 2008 (UTC)

General definition and computation

Doesn't the field K need to be commutative for the determinant to make sense? Dharma6662000 (talk) 18:53, 15 August 2008 (UTC)

Fields are commutative by definition. Algebraist 01:03, 22 August 2008 (UTC)

too hard to understand

i consider myself an every day joe (who completed higher level calculus at high school and did reasonably well) but this page seems to be a reference for experts - it does very little to enlighten the mortals amongst us who are ignorant about matrices and determinants etc. —Preceding unsigned comment added by 122.107.239.188 (talk) 12:48, 17 August 2008 (UTC)

Dubious statement in article-bit complexity of Gaussian elimination polynomial, not exponential

The article dubiously states "What is not often discussed is the so-called "bit complexity" of the problem, i.e. how many bits of accuracy you need to store for intermediate values. For example, using Gaussian elimination, you can reduce the matrix to upper triangular form, then multiply the main diagonal to get the determinant (this is essentially a special case of the LU decomposition as above), but a quick calculation will show that the bit size of intermediate values could potentially become exponential."

[1] proves otherwise-that every number occurring in Gaussian Elimination can be represented as a fraction both the numerator and denominator of which are a polynomial number of bits.

Kstueve (talk) 23:55, 4 December 2008 (UTC)

Mistake in "block matrices" section

I believe there's a mistake in the "Further properties"/"block matrices" section. At the end of the section it claims that the determinant of a block matrix where all the blocks are diagonal is equal to the determinant of the matrix where the individual blocks have been replaced by their determinants.

A simple example shows this is incorrect:

$\det {\begin{pmatrix}2&0&1&0\\0&2&0&1\\1&0&2&0\\0&1&0&2\end{pmatrix}}=9\neq 15=\det {\begin{pmatrix}4&1\\1&4\end{pmatrix}}$

Removed. The real result is in the source cited, if anyone feels like thinking of a good way of summarizing it. Algebraist 20:33, 23 March 2009 (UTC)

And the source cited is [2], if anyone much later from today wonders what it is :) Shreevatsa (talk) 21:00, 23 March 2009 (UTC)

Another mistake

I removed the text:

Similarly in the case that B,C are square (nxn, mxm resp.) and A,D rectangular (nxm, mxn resp.) one arrives at the identities

$\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det {\begin{pmatrix}0&B\\I&D\end{pmatrix}}{\begin{pmatrix}C-DB^{-1}A&0\\B^{-1}A&I\end{pmatrix}}=-\det(B)\det(C-DB^{-1}A)$ if B is invertible and

$\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det {\begin{pmatrix}A&I\\C&0\end{pmatrix}}{\begin{pmatrix}I&C^{-1}D\\0&B-AC^{-1}D\end{pmatrix}}=-\det(C)\det(B-AC^{-1}D)$ if C is invertible.

because the formulae give the wrong answer for

$B=C=\left({\begin{array}{cc}1&0\\0&1\end{array}}\right),~~~A=D=\left({\begin{array}{cc}0&0\\0&0\end{array}}\right)~.$ —Preceding unsigned comment added by Quantling (talk • contribs) 20:07, 19 November 2009 (UTC)

Added new method for N-by-N matrices

I added a reference to the Laplace expansion method under "n-by-n" matrices. However, for now it is a oneliner, would it be ok for me to copy text and pictures from the mentioned page, and put it here to expand it? Or is that overkill and/or illegal? Rumblethunder (talk) 16:21, 24 March 2009 (UTC)

It would certainly not be illegal. However, the text does mention this some paragraphs later. I think having the definition somehow clear-cut is of benefit, so I'd prefer the Laplace gadget in the properties section (instead of an alternative definitions). But anyways, this article needs work... Jakob.scholbach (talk) 17:02, 24 March 2009 (UTC)

I believe it is better to mention it in the same section as the Leibniz method, I think that is the first place people will be looking. Also, I am quite new to Wikipedia, so I am not too familiar with "gadgets" and "properties" in this context...how do you mean? Rumblethunder (talk) 19:46, 24 March 2009 (UTC)

Do you think so? I have no strong preference, but I don't think so. I simply meant to let the Laplace expansion where it is (in the section called "Properties characterizing the determinant"). But if you feel your way is better, go ahead and put the information to the section "Definition". In that case, you should move the text "It is also possible to expand a determinant along a row or column using Laplace's formula, which is efficient for relatively small matrices ..." up to that section. Jakob.scholbach (talk) 20:40, 24 March 2009 (UTC)

I agree with Jakob that there should be only one definition. However, Rumblethunder does have a point that the Laplace formula appears quite late in the article. So, I replaced the sentence that Rumblethunder added with a sentence refering to the later section. However, I think the recursive definition using Laplace's formula is easier to understand than the definition as a sum over perturbations (it's also how I was initially taught determinants, for what it's worth). So there is a case for using Laplace's formula as definition and to move the current definition further down. -- Jitse Niesen (talk) 21:08, 24 March 2009 (UTC)

Nice edit Jitse. I absolutely agree concerning the comprehensibility of the Laplace formula contra other methods, that is also what I was taught. I believe this method should be easy to find for people who are not absolutely conversant with matrices and mathematical notation (the Leibniz method in formal mathematical notation will probably be worse than muddy for most people).Rumblethunder (talk) 21:20, 24 March 2009 (UTC)

Leibniz Formula, Levi-Civita formalism

In the section on the Leibniz formula, I find the explanation confusing, in that it is not obvious (to me at least) what "A1,σ(1) · A2,σ(2) · ... · An,σ(n)" actually means. At least one illustration would help.

Also, it seems to me that showing the Levi-Civita method with appropriate summation convention would help, particularly for those those of us trying to understand the determinant of the GR metric tensor and its transformation properties. LAncienne (talk) 15:09, 9 June 2009 (UTC)

Go ahead! Jakob.scholbach (talk) 19:09, 9 June 2009 (UTC)

Cleanup and 2 x 2

I've tried to clean up the beginning of this article. There is material in the section on 2 x 2 matrices relating to parallelograms that seems inappropriate to me. A simple statement relating the area of a parallelogram to the determinant of the corresponding matrix should suffice here. Am I missing something?--agr (talk) 22:38, 10 July 2009 (UTC)

I'm missing something about matrix-parallelogram correspondence. Please explain. As for the assertion in the lede, you might consider the indicator function for the finite-area region, then review change of variable where the determinant arises for a linear map.Rgdboer (talk) 23:09, 10 July 2009 (UTC)

Relationship to trace

Could somebody add in a few details about proving the relationship to trace formula? I can't figure it out. —Preceding unsigned comment added by 128.250.54.167 (talk) 11:53, 4 August 2009 (UTC)

I tried to clarify this in the article. I don't think we should give all the details in the article, but if you're still having problems with it, let me know and I'll try to explain it here. -- Jitse Niesen (talk) 14:30, 4 August 2009 (UTC)

I added a proof outline that shows how det(A)=exp(tr(log(X))) can be used to prove the "Newton's identities" formulae. I hope it is not too much math for this page. Quantling (talk) 15:08, 5 August 2009 (UTC)

Distributive law

According to http://mathworld.wolfram.com/Determinant.html determinants are distributive. Shouldn't this be mentioned in the article? Or is Wolfram wrong 212.41.92.84 (talk) 08:24, 5 August 2009 (UTC)

Wolfram is right. This attribute is mentioned in the article but it called "multiplicative map." I modified the article to include the word "distributive" as well. Quantling (talk) 15:14, 5 August 2009 (UTC)

Despite what Mathworld says, I don't think the word "distributive" is appropriate in this context. The determinant is not a binary operation interacting with addition. The terms "multiplicative map" and "homomorphism" are appropriate, not "distributive." Does anyone have a truly authoritative source saying otherwise? -- Spireguy (talk) 20:30, 5 August 2009 (UTC)

I was unclear. I meant primarily that Wolfam is right that determinants have that property; I feel less strongly about what the property should be called. I made the edit to the article because Wolfram, a well-known source, uses the word "distributive" and it does seem plausible to me that "determinant is distributive over multiplication." But I am by no means married to that wording. Feel free to remove "distributive" or, perhaps, somehow note that the term is used by this well-known source but may not be completely accurate. Quantling (talk) 20:52, 10 August 2009 (UTC)

I changed the article to indicate that "multiplicative map" is "also termed 'distributive' by Wolfram MathWorld" with the hope that that maximizes relevant information without compromising accuracy. Quantling (talk) 15:07, 13 August 2009 (UTC)

Cleanup suggestion

"Examples" section looks terrible on a 1024xY screen. Not sure how to fix it. Campus IP; don't assume I'm another user of it. 204.102.214.15 (talk) 18:24, 21 September 2009 (UTC)

Abstract formulation - back to previous one?

I don't like the new version of the section "Abstract formulation" by Vincent Semeria, January 2010. The previous one, focus on the wedge product, was more helpul to me. But I'm afraid I don't have the authority to undo those changes. Anybody? Wyoagafos (talk) 11:10, 8 February 2010 (UTC)

LU decomposition

I know that the Leibniz method introduces the concept of parity of permutations just above it, but LU decomposition is simpler to understand than LUP decomposition, so it may pay off to change the section (I am struggling with the parity business, so I found the LU decomposition page actually informative). The fact that the determinant of U is the product of the diagonal elements (for any dimension) is intuitive for people with advanced knowledge of determinants, for those who have learnt stuff by reading the wikipedia article itself (=me), it isn't (the LU decomposition page does though). --Squidonius (talk) 06:22, 8 October 2010 (UTC)

archive

I've set up an automated archiving for this talk page. Jakob.scholbach (talk) 20:43, 27 March 2011 (UTC)

Opening sentence

Is "special" a mathematical term outside my experience? If so, should there be an article about it, with a hyperlink. Otherwise, what is its force in this sentence. In any case, about 20 books I have at hand define a determinant as a SQUARE ARRAY with the value given by the sum of products formula, NOT as a number. If a determinant is defined as a number, how can all the manipulations on rows and columns be discussed? They have been lost in the evaluation. The rest of the lede certainly gives prominence to topics that I have never seen mentioned in an introduction to the topic. I have to stop here -- but the rest of the article is rather disjoint from customary accounts. Might it be a good idea for someone to go through some standard textbook discussions and follow standard coverage? Michael P. Barnett (talk) 02:50, 1 May 2011 (UTC)

The determinant is a number, not an array. It is a property of the array, and row and column operations on the array will affect the value of the determinant, but the determinant itself is a number (a member of the underlying field or ring). The lede seems reasonable to me. I agree that the article feels a bit disjointed, the result of many hands over time, and it could used a general edit to integrate the pieces with a common style; but the content covers the usual standard topics. There is plenty more that could be added, as noted in the section above, though whether all that material belongs in one article is perhaps something that should be discussed. Anything beyond that you would like to add? -- Elphion (talk) 13:50, 1 May 2011 (UTC)

Trying to understand how your many books speak of the determinant as an array, I should mention that there is the usual conflation of the operator and the things it produces. For example, the absolute value of a real number is real number; but there is also the absolute value function on the reals that returns the absolute value of the argument. Similarly, the determinant of a matrix is a number, but there is also the determinant operator that takes arrays into the underlying ring of numbers. -- Elphion (talk) 15:05, 1 May 2011 (UTC)

Many thanks for your comment. I was posting a further comment that hit an edit conflict with yours. Herewith a slight amendation of what I had written. I assume that you have formal training in mathematics. My degrees are in chemistry, but I have picked up some awareness of mathematical topics over the years. Since posting my comment last night, I had checked further in books I have at hand, and found the definition as a number in authoritative sources -- Rektorys, Itô, and others. Unfortunately I had gone first to Mathematics of Physics and Chemistry that I used to teach myself about determinants in 1948+. Sorry. Also I have got into the habit of representing the array that defines a determinant as det[...] in Mathematica sessions, replacing det by Det when I want automatic evaluation (which is seldom).

I got into the matrix element and determinant articles yesterday because I could not remember whether a matrix is Hermitian if or iff it is self-adjoint, and was feeling a bit bruised by WK articles beyond my understanding. It would be presumptuous for me at this time to suggest significant changes to the lede for Determinants that has been gone over by people with far more mathematical knowledge than me. But I hope it is not presumptuous if I mention how I think it might have been started, had nothing been written already. There was an article in the Guardian Weekly last week that reported concern by WK that so few academics contribute to WK articles in their respective professional fields. Abstract algebra is NOT mine, but I have taught math literacy extensively, to students with a wide range of comprehension (including graduate humanists -- answering their questions was extremely beneficial to my understanding).

Back to the lede

1. Suppose someone, Pat Doe, who can handle elementary algebraic manipulation, but has never encountered sets and is afraid of geometry, wants to learn about determinants. I assume that the opening sentences of a lede should be written in a way that avoids possible misunderstanding by someone working in isolation, without a tutor to turn to. I have looked at the present lede with an eye to how it might be misunderstood. Could the dimensionality 2 be considered "an important number associated with any square matrix"? Could the rank be considered an important number associated with it? The trouble is with the word "any". If you have only taught elite students, you may not realize how things that are obvious to you are not necessarily obvious to others. I belong to the "ability to misunderstand" contingent, and empathize.

2. The rest of the opening paragraph will put off a Pat Doe who follows the links, which require more and more mathematical background.

3. The article does seem a bit slanted to the geometrical, relative to all the material I have at hand. But my personal library is a bit outdated, and I cannot get to a major library for several days.

4. Hence my following effort, which I recognize will go no further, but can do no harm.

Michael P. Barnett (talk) 15:52, 1 May 2011 (UTC)

(outdent) For clarity I've taken the liberty of duplicating your signature from below to apply to the remarks above, since the intervening section head guarantees that they will get separated.

I sympathize entirely with your bruising experience in the math articles. Unfortunately, the professional touch often contributes impenetrability rather than lucidity. There are several problems here. In the first place, WP is an encyclopedia, not a text book, and striking the proper balance in presentation is not easy. EB has this problem as well; several of their technical articles jump right in to very technical discussion. And professional mathematicians will overlook things that need explaining: your mention of simultaneous equations below is a glaring omission from the current lede (especially given the history of determinants), because every mathematician will "just know" that that's included in "linear algebra".

In WP the problem is compounded by frequent editing by different hands: the undergraduate throws in a favorite example, and the professional decides the discussion is not suitably general. Both are often useful additions, but without care the organization and readability of the article will suffer. It is hard and time consuming to present technically involved concepts lucidly in the first place, and the will and energy to keep them lucid over time can be hard to maintain. And on taking a closer look at the current article, I will revise my assessment above: it's more than "a bit" disjointed; it needs a thorough overhaul.

There are some good points in your suggested lede below, but I have some reservations, appended below.

-- Elphion (talk) 17:53, 1 May 2011 (UTC)

An alternative opening

In algebra, the determinant of a square matrix (that is, a square array) of numbers is a single number that is computed from the elements of the array by a simple rule that the following example illustrates. The determinant: ${\begin{vmatrix}1&2\\3&4\end{vmatrix}}\$ has the value 1×4-2×3=-2. The rule converts a 3×3 determinant to the sum of 3!=6 terms. Each of these is the product of 3 elements of the matrix. In general, the value of an n×n determinant consists of n! terms. Each of these is the product of n elements of the matrix, selected in a special way. Many kinds of mathematical expression can be used as elements of a matrix, and the corresponding determinants are computed by the rules that are used when the elements are numbers. Determinants are used in calculations throughout engineering and the natural and social sciences. They are particularly important in the solution of simultaneous equations. Determinants have many properties that simplify their computation, which becomes enormously time consuming when the determinants are large. The properties of certain determinants can be explained geometrically. Determinants that consist of other kinds of mathematical objects are used in advanced mathematical and scientific theories.

(further paragraphs appropriate to successive sections of article) Michael P. Barnett (talk) 15:52, 1 May 2011 (UTC)

A good start. Some reservations:

1. The "simple" rule is simple only when n < 3. I think it's a mistake to get into computation (even by example) in the lede. I would like to see a separate section in the article on computation, clearly labeled "Computation of determinants", and the n! number of terms should be explicitly mentioned there.

2. "Many kinds of mathematical expression can be used as elements of a matrix, and the corresponding determinants are computed by the rules that are used when the elements are numbers." I see what you're getting at, but it needs to be stated more carefully. (One could, I suppose, speak of matrices over function rings, but that's not necessarily helpful to the layman!)

3. Similarly, "Determinants that consist of other kinds of mathematical objects are used in advanced mathematical and scientific theories." -- Remarks similar to 2. Also, "advanced mathematics and science" suffices, without the theories.

4. "The properties of certain determinants can be explained geometrically" doesn't really convey much without an example. I think the example currently in the lede might as well stay (and it's a good example); otherwise the geometric aspect probably ought to be eliminated from the lede.

-- Elphion (talk) 17:53, 1 May 2011 (UTC)

The new lede and general issues it raises

Many thanks for rewriting the lede the way you have. It has given me fresh insights to a number of matters. But I wonder who else it can benefit. Which leads me ask, for whom are the mathematics articles in WK intended -- readers or Editors and, if readers, with what range of knowledge? Should the WK article on determinants be readable by someone who can understand explanations of determinants in introductory texts and in reference works that are in many public libraries?

The lede is now very different from the opening of articles about determinants in every published encyclopedic work on mathematics and from every standard textbook and monograph that deals in all or in part with linear algebra that I have inspected regarding the topic. Some are introductory, some are advanced. Perhaps the most institutionally authoritative is the Encyclopedic Dictionary of Mathematics of the Mathematical Society of Japan, that has forewards by the President of that society and the President of the American Mathematical Society at the time of publication. I can provide a full list.

Of all these works, only the text by Poole mentions the geometrical interpretation, and that is in the context of the customary treatment, and diluted by numerous conventional examples.

I can provide simple statistics on the vast number of papers that use determinants because of their non-geometrical properties. What is the evidence for correspondingly massive use of geometrical properties?

I recognize that a strength of WK is potential for updates that provide currency. But does putting the geometry first gibe with widespread current literature and teaching? Is there any need for verifiability that a non traditional approach has been adopted to a major extent, to justify mentioning it first in the lede?

Scholarly papers published in peer reviewed journals often give references to sources of background information, often citing two or three that are typical and consistent. It would be helpful for people increasingly dependent on internet access for WK articles to conform. I could not do this with the Determinant article or with many other WK articles on math and science because they are so inconsistent with outside world presentations.

The impact of the opening sentence of a lede is extremely important. Someone with knowledge of determinants, seeking to refresh or check some details, might well give up after the opening paragraph, because it is so disjoint with what they remember.

Someone who is a bit more tenuous and who is perplexed, as I was, because they do not realize that the word "volume" is being used in the abstract n-dimensional sense, and read on, might become even more concerned. Just follow the link to measure. Then check the following statements.

1 (opening of second paragraph) "To qualify as a measure (see Definition below), a function that assigns a non-negative real number or +∞ to a set's subsets must satisfy a few conditions. One important condition is countable additivity."

A reader who has absorbed the comment in Determinant that sign shows orientation might give up at this point and assume the author of the new Determinant lede is abusing terminology.

However, a reader who goes on will find, under Measure (mathematics)#Generalizations:

2. "For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity."

So why the restriction in the first place, and what needs to be satisfied for this generalization to be safe (in sense of not opening a route to fallacies)? The reader might give up trying to use WK math articles at that point because they seem to lack the rigour that should attend the terminology that is being used. Worse, if the reader goes on for about another 15 lines and finds

3. "The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer."

Not so OK -- a very clear indication that a WK writer disagrees with outside sources, raising concern about who to believe. And the word "mysterious" really jars in mathematical discourse that to someone with my limited knowledge seems unnecessarily eclectic. This can engender a scepticism about WK mathematics at large.

However, if the reader guessed that "measure of volume" in two dimensions means "area" (in its elementary school usage), and wants confirmation, the cursory Google search I just ran did not help.

So, for a topic that dozens of textbooks and encyclopedic works explain concisely and readably, WK just takes the time of a tenacious reader without providing help.

Am I out of order on this? If not, is the situation remediable? Michael P. Barnett (talk) 17:37, 2 May 2011 (UTC)

OK, several points. This response is not meant as negative criticism, and as I indicated above, the article does need work. Given my advice below to complain on talk pages, I think it's healthy that you're complaining on the talk page!

Just to be clear: the "new" lede is hardly new -- I only cleaned up the language a bit. I hope I haven't misled you to anticipate a major rewrite on my part any time soon; I haven't the time.

But trying to get me to agree that the geometric aspect of deteriminants is not a key point is a losing battle. This is covered it most modern introductory linear algebra texts, for the simple reason that it is key to understanding the geometry of linear transformations and their use in geometric applications like the calculus. If anything, I think this is not adquately reflected in the current article. Whether a transformation collapses the domain (det = 0) is important in analyzing simultaneous equations (and as I said above, the application to simultaneous equations does deserve explicit mention in the lede), but that's hardly the whole story. And the lede clearly states that the absolute value of the determinant deals with measure. Negative volume is never suggested.

Your complaints about the article Measure (mathematics) should be taken there, where editors can see them. That that article has its own issues is not a good reason for not linking to it, and not a good reason for rehashing the topic here for the points relevant to determinants. That way madness lies; we have to rely on links or the articles will expand beyond control.

Of course you are right that an inline judgment like "a misnomer" should be referenced. But its presence doesn't dismay me. It is a good signal that I should be wary of taking the term too literally. It would be better if the editor had explained why it's a misnomer. But the math articles are not like, say, the articles on TV series: that an editor wrote that means there's likely something to it. An appropriate response is to tag it with {{cn}}.

And it certainly doesn't bother me that this "can engender a scepticism about WK mathematics at large" -- *any* WP article should be read skeptically; nothing here should be taken on blind faith. You must evaluate the information as presented, and if you find it wanting, complain on the talk page. You seem to be asking for a grand unified approach to all the math articles, but as I implied above, such a thing is not likely to happen. The field is too vast. The best we can hope for is a collection of information that a large number of people find useful, and that gets improved when people don't find it so.

Your suggestion of listing further reading for all math articles has merit. You might want to take that to Wikipedia talk:WikiProject Mathematics.

-- Elphion (talk) 21:57, 2 May 2011 (UTC)

Thanks to you two for working on the lead. I also have this article on my todo-list (so far, didn't get much further than cleaning up some parts and creating the above sketch of possible topics). Before you engage in lengthy discussions about the lead, though, I'd like to kindly suggest to work on the article first. The lead is important, but generally changes a lot when the article changes. I often found it helpful first having a sound and stable article, then working on the lead. (Otherwise the time and energy you are spending on the lead now might not be used 100% effectively, since it might need to be reworked later anyway.)

I hope to join in once I'm done with the Logarithm FAC. Sigh... Jakob.scholbach (talk) 22:04, 2 May 2011 (UTC)

Thanks for the advice. But to clarify, I did not hope to convince anyone of anything, just wanted to test the waters of whether I should try to be responsive to recent newspaper article about the concern of WK about shortage of academics who contribute from their field of expertise and see if the perspective based on my experience could help (teaching, research, publication). I am not offended at being written off as a fossil. I am 82. Whatever does or does not go in the lede, it seems clear that NPOV as used here weighs heavily against readers who are concerned with traditional use of determinants, that references are not needed outside the history section, and readers have to know when words that are commonplace, e.g. "volume", are used with technical meaning, and words that look a bit technical, e.g. "characteristic number" do not. Michael P. Barnett (talk) 01:01, 3 May 2011 (UTC)

Welcome to WP. And yes, we can always use people with a scientific background (and also people without). The better the articles become, the more knowledge and care is needed to improve them. Hope you stay around--WP:WPM is generally a quite nice atmosphere. Again, welcome!

Some historical comments

Infinite determinants are of considerable importance in planetary theory aka celestial mechanics from 19th century on. It is significant that Modern Analysis by Whittaker and Watson, which marked a major development in mathematics, certainly in England, has substantive section on infinite determinants (2.8, pp 36-37). Infinite determinants with a particular systematic structure, truncated for numerical work, were the basis of the calculation of electronic structure of 2 electron atoms by Pekeris -- outside mainstream of atomic energy calculations, but extremely important at time, providing theoretical results comparable in accuracy with experimental. Quickest way for me to give references to several relevant papers is bibliography of Barnett, Decker and Krandick, J Chem Phys 114 23 10265 2011. (This introduced further computational tactics, but I am not pushing for its mention). Michael P. Barnett (talk) 21:02, 3 May 2011 (UTC)

Observing Wiki guidelines

1. If successive sentences in the lede are supposed to indicate content of successive sections of article, a new first section should be provided or the first paragraph of present lede should be deleted. Or the present lede rearranged and a geometrical section put later in the article.

There is no strict rule on this, but surely the lead should roughly match the article. In this case, I don's see why you want to delete a part of the lead. Section 1.1. does mention the parallelogram interpretation of det of 2x2 matrices. Jakob.scholbach (talk) 09:52, 4 May 2011 (UTC)

Also there are sections "Volume" and "Jacobian determinant" under "Applications". I am personally not very happy with the emphasis on the volume interpretation because it is not very general (quite hard to give any clear meaning to it unless entries are real numbers) and in fact not very elementary either; the only reason for mentioning it early is that it is intuitive (which general determinants maybe are not). I did modify the sentence in the past (notably stressing real numbers) and am responsible (I think) for the "measure of volume" phrase, which is admittedly not very beautiful. However saying either just measure or just volume immediately leads to the question "of what?". Also it is unfortunate that volume means area in dimension 2, but this is common for geometric terminology in varying dimension ("space", "parallelepiped", "hyperplane") that is often anchored in dimension 3. Marc van Leeuwen (talk) 12:03, 4 May 2011 (UTC)

2. If a definition is supposed to have a verifiable citation attached, I think the geometrical definition should be close enough to a published definition to avoid the "original thought" that seems to have gone into the present wording, and the citation should be provided.

In the guise of the area of the parallelogram or volume of the parallelepiped, the material is standard. Of course, a reference for this cannot hurt. Jakob.scholbach (talk) 09:52, 4 May 2011 (UTC)

3. In the rest of the article, a few more citations seem necessary. Unless WK guidelines are waived for articles on math. Michael P. Barnett (talk) 21:02, 3 May 2011 (UTC)

True. In case you don't know: WP has a number of citation templates which make the references look a little more smooth and standardized. For this and more, see Wikipedia:WikiProject_Mathematics/Reference_resources. To format the {{Citation}} templates, I once wrote a database (called zeteo, see there, too). Jakob.scholbach (talk) 09:52, 4 May 2011 (UTC)

Additive rule for determinants

I've removed the following section, recently added by 129.67.40.84 (talk). The reason is that this is clearly recent research, as far as I can see not yet published in a peer-reviewed journal, the reference is to transparencies of a presentation that in themselves do not provide sufficient context to understand the statement fully, and that the text contributed to the WP article is taken from this presentation unmodified. While the material could be interesting, I deem it premature for inclusion in the determinant article, more so because it could easily confuse readers looking for a simple formula for the determinant of a sum of two matrices (which does not exist, note the condition $N\geq n+1$ that says this only applies to sufficiently large sums of determinants). Of course it could be inserted in the article once the situation is sufficiently improved. Marc van Leeuwen (talk) 07:05, 6 May 2011 (UTC)

I inserted the section, perhaps too quickly, because the result is quite interesting, and seams to fit well into the article in general. I've tried to clean it up some, and to help people avoid the trap you rightfully mentioned. On the overall correctness of the theorem I have tested it for many values of N and n, so at least it is not obviously false. --Thomasda (talk) 17:32, 9 May 2011 (UTC)

Is this published somewhere (other than slides of a talk)? If no, we cannot take it. If yes, we can in principle take it, but even then I think we should devote at most one sentence to this theme. Before doing that, we should check whether this theorem is notable, possibly by looking for papers that cite this given paper. Jakob.scholbach (talk) 21:48, 9 May 2011 (UTC)

It's published a few places. The latest version seams to be this one at arxiv.org. I don't know how to find out if it is notable, but the WK article contains a lot of stuff on properties for calculation of determinants. As Marc said, many people probably come wondering if there is an addition formula as well. Since it is such a simple question to ask, it would be nice to have a discussion of what is possible and not possible in that regard. --Thomasda (talk) 13:29, 10 May 2011 (UTC)

By publish I did not mean preprints such as arxiv, nor that someone uploads it on some web page. We need to refer to peer-reviewed scientific journals. Is this published in any such journal? Jakob.scholbach (talk) 14:35, 10 May 2011 (UTC)

Is this result really notable on its own merits? It doesn't really qualify as a computational device: the RHS is significantly more complicated than the LHS, involving a whole boatload of matrices no smaller than the original. It would be simpler to add the matrices (trivial) and compute the determinant once. And as the discussion above indicates, this article has more pressing needs than to indicate why this result deserves space. Elphion (talk) 15:13, 10 May 2011 (UTC)

Determinant of Sums of Matrices

Given enough matrices, the determinant of their sum can be found using an Inclusion–exclusion principle like formula. More precisely, the number of matrices summed must be greater than the size of each matrix. Notice this means, that this doesn't work for $det(A_{1}+A_{2})$ unless $A_{1}$ and $A_{2}$ are 1x1 matrices.

The theorem says, that "given $A\in M_{n}$ and an integer $N$, with $N\geq n+1$. For any $N$-tuple $S=(A_{1},A_{2},...,A_{N})$, $A_{i}\in M_{n}$, $i=1,...,N$, the following relation holds:"

$\sum _{k=0}^{N}(-1)^{k}\sum _{\Omega \in \Sigma (S),|\Omega |=k}det(A+\sum _{A_{i}\in \Omega }A_{i})=0$.

Here, $|\Omega |=k$ means that $\Omega$ is a formal sum with $k$ summands, and that $A_{i}\in \Omega$ means that $A_{i}$ is a summand in $\Omega$.

For example, that theorem tells us, that for matrices of size 1x1, 2x2 or 3x3 the following equation holds:

${\begin{aligned}det(A_{1}+A_{2}+A_{3}+A_{4})&=det(A_{1}+A_{2}+A_{3})+det(A_{1}+A_{2}+A_{4})+det(A_{1}+A_{3}+A_{4})+det(A_{2}+A_{3}+A_{4})\\&-det(A_{1}+A_{2})-det(A_{1}+A_{3})-det(A_{1}+A_{4})-det(A_{2}+A_{3})-det(A_{2}+A_{4})-det(A_{3}+A_{4})\\&+det(A_{1})+det(A_{2})+det(A_{3})+det(A_{4})\end{aligned}}$

Again it is worth pointing out, that the formula works only for large sums of matrices. If we try to use it to calculate the determinant of just two matrices, we'll get an error:

$\left|{\begin{array}{cc}a+\alpha &b+\beta \\d+\delta &e+\epsilon \end{array}}\right|-\left|{\begin{array}{cc}a&b\\d&e\end{array}}\right|-\left|{\begin{array}{cc}\alpha &\beta \\\delta &\epsilon \end{array}}\right|=e\alpha +a\epsilon -d\beta -b\delta$

Removed tag at top

I have removed the tag at the top after coming here from WT:V and looking at what the tag was about. There seems to be no major issue, just a mix up of how something is represented sometimes with what it is. Dmcq (talk) 08:37, 14 May 2011 (UTC)

Discussions elsewhere

There have been extensive discussions regarding this article many places elsewhere including my talk page, Michael's talk page and wp:ver. . IMHO Michael P. Barnett exhibits so much caution and politeness and interprets feedback in that same manner that he read it that he had been shut down here. I think he's ready to come back, if so please understand this situation. North8000 (talk) 11:19, 19 May 2011 (UTC)

Goodness. I had no idea the exchanges above had led to such a firestorm. We could have had a more productive time with more light and less heat had Michael kept the discussion here, since this is the article he wants to improve. I agree whole-heartedly with your remarks at WT:V, and I hope Michael doesn't feel that we are trying to squelch discussion or edits.

My impression is that the main difficulty with the lede is that it does not clearly distinguish between the determinant of a matrix and the determinant operator det:S^n×n → S. I think it would be a mistake to try to give a rigorous definition of the det operator in the lede. Something like "For a given commutative ring S and non-negative integer n the determinant operator det:(Sⁿ)ⁿ → S is the essentially unique n-ary skew-symmetric operator with the following properties ... " would scare everyone away instantly. It makes more sense to start with the determinant of a matrix, and lead into the determinant operator mapping matrices to numbers. Something like:

In algebra, the determinant of a square matrix with entries from a number structure S (a field like the real numbers, or more generally a commutative ring) is a number from the same underlying structure S. The determinant operator det that maps n×n matrices over S into values from S does so in a manner that preserves multiplication — det(AB) = det(A)·det(B) — and such that a matrix A is invertible over S if and only if det(A) is an invertible member of S.

Already this is arguably too general. It's hard to introduce det with suitable generality right off the bat, and yet common examples like the Jacobean are already technically determinants over rings, not fields. Perhaps it makes sense to start even more specifically: "The determinant of a square matrix over the real numbers is a real number, assigned in such a way that det(AB) = det(A)·det(B), and such that A is invertible if and only if det(A) is nonzero. More generally, determinants can be defined for matrices over other fields, like the complex numbers, or even over an arbitrary commutative ring (where A is invertible if and only if det(A) is an invertible member of the ring)." After that it would make sense to observe that the determinant operator can be seen as an n-ary skew-symmetric operator etc. etc. The main article needs a section early on defining the determinant operator this way.

I agree with Michael that the geometric interpretation doesn't belong in the next breath of the lede, but I do believe it deserves an example in a subsequent paragraph somewhere in the lede.

-- Elphion (talk) 18:12, 19 May 2011 (UTC)

Thanks. My main thought is that you are dealing with someone who is unusually cautious and polite and to please keep that in mind. Sincerely, North8000 (talk) 20:04, 19 May 2011 (UTC)

Suggestions for added content

This replies to Dmcq's most recent comment, but I thought it better to start a new section.

Symbolic calculation of determinants

Are we referring to the same Markov algorithm? I mean the algorithm that is used in symbolic calculation software to construct symbolic expressions of symbolic determinants. Consider a determinant of order n, that contains elements a + b x, where a and b stand for real numbers and x stand for a symbol. Expansion gives a polynomial of degree n in x. Compute the numerical values of the determinant with x replaced by 0, 1, ..., n-1. Take differences. Hence the coefficients in the characteristic polynomial. This is described in K.O.Geddes, S.R.Czapor and G.Labahn, Algorithms for computer algebra, Kluwer, 1992. By extension, I worked with determinants containing elements a + b x + c Z. I needed to push the order of the determinant as far as possible in a list that included 308. This was the highest (in the list) that the system could handle when each element contained at most one variable. So my co-workers constructed the characteristic polynomials in x for Z=0,1,2,... and we took differences, to construct a characteristic polynomial in x, in which coefficients were polynomials in Z (as an explicit symbol). Faddeyev and Faddeyeva refer to this as the Markov algorithm, in Computational Methods of Linear Algebra, tr. R. C. Williams, Freeman, 1963. In fact, it was invented prior to Markov's formulation by B. L. Hicks, Journal of Chemical Physics, 8, 569, 1940. Hicks work was described in the standard text on theory of infra-red spectra by Wilson, Decius and Cross and it was the preferred method for constructing characteristic polynomials in theoretical chemistry for about 20 years. In suggesting mention of the general method I did not imply referencing how I used it.

Infinite determinants and differential and algebraic equations

Are we talking about the same infinite determinants? I learned about these during a brief encounter with the Mathieu equation about 60 years ago that led me to Hill's work on lunar theory and to Section 2.8 of Whittaker and Watson. Active research and publication continues on these interrelated topics, e.g. Curtis Wilson, The Hill-Brown Theory of the Moon's Motion, Springer, 2010; L. Wille and R. Phariseau, On the computation of a certain class of Hill determinants, Journal of Computational and Applied Mathematics, 15, (1) 83-91, 1986; I can run SCOPUS and Web Of Science searches on this if need be. Whittaker and Watson mention Fürstenau's work on algebraic equations, and the continuants of Sylvester. The "hand waving" explanation, that I have used in seminars for how infinite determinants turn up in calculations of planetary theory and quantum mechanics is by extension of the Frobenius method that expands a differential equation in a series of special functions (e.g. Laguerre polynomials, and establishes a recurrence relation between successive coefficients. In this approach to the "helium problem", (one of the most famous achievements of Chaim Pekeris), this contains 33 terms, that overlap from equation to equation, giving an infinite set of simultaneous equations in the coefficients. These contain the variables I labeled x in the preceding paragraph.

Determinants and bilinear forms

I remember looking at Muir's monograph about 20 years ago, apropos the "helium problem", and finding material on bilinear forms that I experimented with, but cannot recall why. So I just ran a quick search for a connection and found a paper that looks quite mathematical: Xuanting Cai, A Gram Determinant for Lickorish's Bilinear Form, [3] indexed under Mathematics>Geometric Topology.

Resultants and polynomial systems

Recent work on Dixon resultants, e.g. Robert Lewis, Heuristics to accelerate the Dixon resultant, Matcom 77 (4) April, 2008 [4] is an important alternative to the Gröbner basis approach to the solution of polynomial systems of major importance in theoretical studies of protein folding and other practical problems (there is an overview of these in Manfred Minimair and Michael P. Barnett, Solving polynomial equations for chemical problems using Gröbner bases, Molecular Physics, 102, (23-24) 2521-2535, 2004.) Might there be some mention and/or further links to some of these topics/people and to bilinear forms and elimination theory and suchlike?

My non-neutral point of view

The chance of anyone who is still following this discussion looking at my joint paper on Gröbner bases is "vanishingly small" (I really am trying to write wiki-ese) but I had better come clean now. The whole point of the paper was to explain the operation of Buchberger's algorithm without falling into the usual style, so eloquently described by the Wigner medalist Harry Lipkin: "As a graduate student in experimental physics ... all attempts to follow a lecture (on certain algebraic topics) resulted in a losing battle with characters, cosets, classes, invariant subgroups, normal divisors and assorted lemmas. It was impossible to learn all the definitions used in one lecture and remember them until the next. The result was complete chaos. It was a great surprise to learn later on the (1) the techniques can be useful, (2) they can be learned without memorizing the large number of definitions and lemmas which frighten the uninitiated." Michael P. Barnett (talk) 19:59, 21 May 2011 (UTC)

No nothing there sounds in the least suitable for this article. They are not about determinants. You should set up separate articles. Dmcq (talk) 20:18, 21 May 2011 (UTC)

I have had a read of what you wrote above and much of it makes no sense to me even though I am familiar with most of the individual concepts like Markov algorithms and the Mathieu functions and Gröbner bases. I am not a professional mathematician so it is quite possible it does make sense but I am pretty certain that if it does it is at a level which has not been reached by most of Wikipedia yet and they haven't made general notability yet. So I don't think they would even be suitable for separate articles. Dmcq (talk) 20:41, 21 May 2011 (UTC)

Sorry. I was trying to be constructive in response to how the article has improved, got carried away with enthusiasm, and it misfired. The less I write now the better. But I would like to remedy one point asap. I should have said "an algorithm attributed to Markov that is completely unrelated to the famous Markov algorithm". The authors of what was, for some time, the standard Russian text on linear algebra use the term "Markov algorithm" for the process I mentioned. Maybe this shows their parochialism (or maybe it predated). I was trying to be terse. Your reaction has been very salutary. I have fallen into the trap of assuming that specialized pieces of information known to me are known to everyone else. Sixty years ago the construction of numerical tables from "differences" was all pervasive in computation. After I posted my message I noticed the "to-do" list. This includes algorithms to compute symbolic determinants! What I tried, but failed, to describe in a simple fashion, is a standard method, which has accessible verifiability. May I have another crack at it in a few days time. Incidentally, I assumed that one of your specializations is abstract algebra. Michael P. Barnett (talk) 21:14, 21 May 2011 (UTC)

Well I can compute the probabilities in Markov processes too but it didn't sound like you meant that either. No maths is just a hobby of mine along with few others. I've given a few talks on subjects that strike my fancy to a local maths society but that's about it. Dmcq (talk) 21:51, 21 May 2011 (UTC)

Notable special cases

There are some well known determinants like the Vandermonde determinant that should be mentioned in this article. Many more examples are given here. Count Iblis (talk) 21:54, 21 May 2011 (UTC)

Just click on Category:Determinants at the bottom of the article and you get a pile of them. Yes the more important should at least be added to the see also list. Dmcq (talk) 21:58, 21 May 2011 (UTC)