Matrix Perutations .. help

[mooeypoo]

Hey guys,

 

I am asking a question about my homework, but my point is to try and get general explanation about the method (since I am quite lost here, and the book isn't very helpful). Which is why I'm posting it here and not in the HW help section. I am using my hw as an example only; if I understand the point I will (hopefully) be able to solve this myself.

 

Okay, then. I started a new advanced physics course (2, actually, expect questions about the math of the other one soon) and there's a lot of math that gets me quite confused. I am familiar with the general principles, but I think that somewhere I'm getting myself confused over the terms and permutations. Help.. please..

 

The question (as an example):

By calculating the components, verify the identity:

[math]\sum_{k}\varepsilon_{ijk}\varepsilon_{mnk} = \delta_{im}\delta_{jn} - \delta_{in}\delta_{jm}[/math]

 

Use this to obtain the simplification of (A x B) \cdot (A x B)

 

Ooookay then. I know that

[math]\delta_{im}=\sum\lambda_{ij}\lambda_{kj}[/math]

Which is the difference between the two angles.

 

Also, C=AxB, and

[math]

C_{i}=\sum_{j,k}\varepsilon_{ijk}A_{j}B_{k}[/math]

 

So this is supposed to help me breaking down the components. Here's what I've tried to do:

 

[math]

(AxB)_{i}=\sum_{j,k}\varepsilon_{ijk}A_{j}B_{k}

(CxD)_{i}=\sum_{m,n}\varepsilon_{imn}A_{m}B_{n}

 

(AxB)\cdot(CxD)=\sum_{i}(\sum_{j,k}\varepsilon_{ijk}A_{j}B_{k})(\sum_{m,n}\varepsilon_{imn}B_{m}D_{n})

[/math]

 

Then, I tried to break them into their components:

[math]

(\sum_{j,k}\varepsilon_{ijk}A_{j}B_{k}) = [/math]

[math]i=1 [/math]

[math]A_{2}B_{3}-A_{3}B_{2}[/math]

[math]i=2[/math]

[math]A_{3}B_{1}-A_{1}B_{3}[/math]

[math]i=3 [/math]

[math]A_{1}B_{2}-A_{2}B_{1}[/math]

And the same with C and D. I have the feeling that the difference between the components lead to the Kronecker Delta (since it's a difference too) but I'm not sure, and I am quite confused with all the symbols and different applications of them.

 

Aaaaand...I have no clue where to go from here.

 

Meh, I think I got completely confused here. Help?

 

thanks in advance,

 

~moo

[ajb]

Advice; Never write out the components in full like that.

 

So, [math](A\times B)_{i} = \epsilon_{ijk}A_{j}B_{k}[/math] where we sum over repeated indices.

 

Also [math]A\cdot B = A_{i}\delta_{ij}B_{j} = A_{i}B_{i}[/math].

 

Thus

 

[math](A\times B)\cdot(A\times B) = \epsilon_{ijk}\epsilon_{imn}A_{j}B_{k}A_{m}B_{n} = (\delta_{jm}\delta_{kn} - \delta_{jn} \delta_{km})A_{j}B_{k}A_{m}B_{n},[/math]

 

which can be written as [math](A\times B)^{2} = A^{2}B^{2} - (A \cdot B)^{2}[/math].

Edited September 7, 2008 by ajb