Talk:Determinant

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--Cronholm¹⁴⁴ 04:01, 16 June 2007 (UTC) Uprated to top priority. Arcfrk 06:12, 28 June 2007 (UTC)

1 Formula
2 Requests
3 Misc
4 Alternate definition?
5 Clarification
6 Mistake in image
7 Using the site in French
8 Sarrus Scheme
9 Diagram: 3D, singular
10 Vertical bar notation
11 Right handed coordinante
12 French Featured Article
13 Discrepancy between image and written material
14 Alternate characterizations
15 My school gave me......
16 Properties (NEW - Check this and delete, if unnecessary)

[edit] Formula

It would be useful to add this formula

$| A + B C D | = | A | | C | | C - 1 + D A - 1 B |$

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)

[edit] 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)

[edit] 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:

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)
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)
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.

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.

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....

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

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.

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) = [b f - c e, c d - a f, a e - b d]

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)

[edit] 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)

[edit] 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)

[edit] Mistake in image

I think there is a mistake in the current version of Determinant.jpg. Please have a look at Image talk:Determinant.jpg. --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)

[edit] 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).

[edit] 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).

[edit] Diagram: 3D, singular

It is requested that a diagram or diagrams be included in this article to improve its quality.
For more information, refer to discussion on this page and/or the listing at Wikipedia:Requested images.

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)

[edit] 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)

[edit] Right handed coordinante

the following sentence is not clear. "The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed." what does "right-handed coordinate system" means? the "coordinate system" article does not mention it. amit man

[edit] 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)

[edit] 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)

[edit] 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)

[edit] My school gave me......

Three math Determinant problems but I don't know how to calculate them they have fractions in them. can someone show me how to figure out this one please?

|5x-4y=7| |x=5-3/2y|

I have been trying different ways to get the answer for instance I tried taking the 2 and making that my x but I think thats wrong and then I tried just leaving them how they were making my(point) answer come up to(-9.5/3.5, -18/3.5) but it does not check out in the equation. So then I I tried the other way again and got the (point) of 2 (1/2)/(3 1/2), 5 (1/2)/(3 1/2). But this answer does not check out either so can someone help me out in figuring out this answer?

Appreciation from Me to whom ever helps.

[edit] Properties (NEW - Check this and delete, if unnecessary)

In properties section, there is this text:

The determinant of a matrix A exhibits the following properties under elementary matrix transformations of A: 1. Exchanging rows or columns multiplies the determinant by −1. 2. Multiplying a row or column by m multiplies the determinant by m. 3. Adding a multiple of a row or column to another leaves the determinant unchanged.

Are you sure that 2 is true, and multiplying a row or column by m doesn't multiply the determinant by 1/m, since the determinant of the new table equals the determinant of the old multiplied by m, so the new table multiplied by 1/m equals the old table...