Talk:Determinant

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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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

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)

<math>B=\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}<math>

which has the determinant (in vector form)

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

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

<math>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}<math>

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)