Question

For Parth:

\(T=\frac{1}{2}\sqrt{\sum_{(a\, b\, c)} \left| \det \left |
\begin{array}{ccc}
x_{a,1}& x_{a,2}& x_{a,3}\\
x_{b,1}& x_{b,2}& x_{b,3}\\
1& 1& 1
\end{array}
\right | \right|^2}.\)

Answer 1

And when you edit the \(\TeX\), refresh the page.

Answer 2

Forgot about refreshing.

Answer 3

This is beautiful; I don't get any of it!

Answer 4

It's a combinatorical sum. It's the real reasoning behind an extremely nasty formula.

Answer 5

By looking at the 3-tuple problem in this venue, you can generalize to n-tuples.

Answer 6

How did you even generalize it?

Answer 7

Oh, sorry for asking. I already know that Limitless the Great is obviously great.

Answer 8

I first found the ugly formula on Wiki. Then, I stared at it for a while and it hit me: This is a combinatorical sum in hiding. That's when I rewrote it as you see above. You won't see this anywhere else on the web.

Answer 9

Also, when I say generalize, I mean that by looking at this combinatorical interpretation of it with respect to summations and determinants, one gets a hint at what the n-tuple version will look like.

Answer 10

I get everything you are doing here!

Now, I just need to learn about matrices.

Answer 11

Bets?

Answer 12

Matrices are simple, Parth.

Answer 13

They are arrays of numbers.

Answer 14

I don't see any good resource on learning about them.

And that I know :_0

Answer 15

They only take on more advanced notions and properties in linear algebra when you analyze their importance in GL_2 and general matrix algebra.

Answer 16

@abb0t I have seen you doing differential equations. =\

Answer 17

John, do you have any more innovations? We could make a team in Google Science Fair.

Answer 18

Abstract/linear algebra are where things get complicated. Even then, it's not that bad. You don't have a lot to learn right now about matrices. Learn the basic operations, the importance of scalars, the eigenvector equation, the determinant rules for 2x2 and 3x3, the generalized summation (even if you can't use it) for n x n determinants, and the aspects of matrix exponentiation along with some other basics.

Answer 19

The eigenvector equation is extremely important.
\(\vec{v}\) and \(\lambda\) are called eigenvectors and eigenvalues if and only if
\[A\vec{v}=\lambda\vec{v}.\]
Note that \(A\in \text{M}_n(\mathbb{R})\), \(\lambda \in \mathbb{R}\), and \(\vec{v}\in \mathbb{R}^n\).

Answer 20

\(\text{M}_n(\mathbb{R})\) is the group (or ring) of n by n square matrices with real entries.

Answer 21

Ah, ah.

Answer 22

So essentially, \(A\) is a just a matrix? And \(\lambda, A\) are scalars, yes?

Answer 23

Be careful. It is very important to pay close attention to the way I described them. \(A\) is a _square_ matrix. \(\lambda\) is a real scalar. \(\vec{v}\) is an _n_ dimensional vector. Notice that n appears in the description of the matrix's dimensions and the vectors: the dimensions must be equal.

Answer 24

The rest of the stuff I mentioned is really easy. You can learn all of that by tomorrow.

Here is something I think is worth knowing, though:

http://en.wikipedia.org/wiki/Matrix_exponential

Answer 25

Not Wiki. Not Wiki.

Answer 26

Wiki is where I learn.

Answer 27

Look where I am now. :-)

Answer 28

Not me.

Answer 29

Ok, fine.
http://mathworld.wolfram.com/MatrixExponential.html

Answer 30

Yeah, yeah, you're genius :-)

Answer 31

NOOOOOOOOOOOOOOOOOOOOOO!!!!!!!!!!!!!!!!!!1

Answer 32

I'd rather take Wiki. -_-

Answer 33

INTERESTING.

Here's something neat! . . .

Answer 34

http://mathworld.wolfram.com/KroneckerProduct.html

Answer 35

I don't have the patience to rewrite all that, but it is pretty awesome.

Answer 36

I don't have the patience to read all that, but it is pretty awesome.

Answer 37

:-)

Answer 38

Phew, I can finally understand what this question means at least.