Transforming Index to Matrix Form: A Guide for Amateurs

[Marin]

Hi everybody!

I have a question concerning tensors and hope you could help me :)

http://www.rzuser.uni-heidelberg.de/~mbernha3/tensoren.pdf

I would like you to look at expressions 13 and 16 :) I hope you won't be bothered by the fact the file is in German. I'm was wandering how these transformations are being made (from the index form to the matrix form) :)

Take in consideration I'm just an amateur :D


Thanks in advance!

Marin

 

[mrandersdk]

Lets look at (13). The a and c on the omega run like an index, that is a=x,y,z , c=x,y,z.
Then let the (x,x) be the first entrance in the matrix, and (y,x) be the next (the one directly below), just like you would say fx. entrance (2,3) now you just use (y,z) instead.

Now let's try to calculate (y,x), that is (remembering summing over repeated index)

$$\Omega^{yx} = \epsilon^{ybx} \omega_b = \sum_{b=x,y,z}\epsilon^{ybx} \omega_b = \epsilon^{yxx} \omega_x + \epsilon^{yyx} \omega_y + \epsilon^{yzx} \omega_z = 0 \omega_x + 0 \omega_y + 1 \omega_z = \omega_z$$

just like that entrance in the matrix, you see?

 

[HallsofIvy]

All they are doing is representing A_ij as the number in the i^th row, j^th column of the matrix.

 

[Marin]

Thanks a lot, I got it :)

It's a little bit of tedious calculations, but I'll try and get these to examples by myself, to get some practice :)

Thanks once again :)

 

[Marin]

There's another question risen up :)

what's the difference between:

$$\Omega^{ac} = \epsilon^{abc}\omega_b$$

$$\Omega_{ac} = \epsilon_{abc}\omega^b$$

$$\Omega_a^c = \epsilon_a^b^c\omega_b$$

$$\Omega^a_c = \epsilon^a_b_c\omega^b$$

I know that the upper indexes are the contavariant, the lower - for the covariant components. But the matrix will always be one and the same. Maybe this has some physical meaning? And there are actually two more combinations to be made: e.g.

is $$\Omega_a^c = \epsilon_a^b^c\omega_b$$

equal to that $$\epsilon_a_b^c\omega^b$$

?

 

[Phrak]

I don't know what the thrust of your text is. But I don't see any references to basis vectors or metrics, which is what raising and lowering indeces is all about. Without the metric concept it must seem a bit of a mystery what sort of objects beyond arrays of numbers are being manipulated. Perhaps that comes in later chapters.

In the mean time, it may help to treat objects like the Levi-Civita tensor $$\epsilon^{ab}\:_c$$ as a matrix that has vectors as elements $$(\epsilon^a)^b\:_c$$.

The basic equation that bridges matrices and tensors is

$$u = Tv$$, that becomes $$u^a = T^a\!_b v^b$$ in tensors.

The column vector v is transformed to the column vector u. This requires that T have a raised index for rows and a lower index for columns.

A strange object like $$U_{ab}$$, that seems have rows in both directions can be viewed as a row vector of row vectors.

 

[Marin]

I apologise for the misunderstanding, I also try to learn LateX and it's a bit of complicated.
So, let's try again typing something:

$$: \\Omega^{ac} = \\epsilon^{abc}\\omega_b$$

$$: \\Omega_{ac} = \\epsilon_{abc}\\omega^b$$

$$: \\Omega_a\\^c = \\epsilon_a\\^b\\^c\\omega_b$$

$$: \\Omega^a\\_c = \\epsilon^a\\_b\\_c\\omega^b$$

are these equations correct and what's the difference between them. If they are correct, is every possible combination of upper and down indexes possible, in order to contract the 'b' abd get the initial tensor?

This part with vectors in the vectors and so on is already clear, but does not explain the physical meaning of Omega in 13 and L in 16 in:

http://www.rzuser.uni-heidelberg.de/...3/tensoren.pdf

which transformed both give normal matrices, despite in the first case both indexes are up (contarvariant) and in the second - both are down (covariant) the tensor.

 

[Marin]

ok, I give up latex, here it is the traditional way: O is omega e is epsilon and o i omega small, so:

O^(ac) = e^(abc)o_b
O_(ac) = e_(abc)o^b
O(_a/^c) = e(_a/^b/^c)o_b
O^(a/_c) = e(^a/_b/_c)o^b

 

[cristo]

ok, I give up latex, here it is the traditional way: O is omega e is epsilon and o i omega small, so:

O^(ac) = e^(abc)o_b
O_(ac) = e_(abc)o^b
O(_a/^c) = e(_a/^b/^c)o_b
O^(a/_c) = e(^a/_b/_c)o^b

I've tidied up post #5 for you. Click on each equation to see the code.

 

[Marin]

cristo, thanks fore retyping the equations above :) So the command for LaTex is just '[tex]' :) ?

Btw, do you have any suggestions to my questions above? I'm pretty interested in tensor algebra, but I find all Wikipedia definitions and expressions a bit out of my current mathematical abilities :(

And another question: Do you happen to know, where I can download Latex from, so that I could practise at home and not make these lame mistakes over here?

with best regards, Marin

 

[Phrak]

Martin-- In Euclidian space in Cartesian coordinates, upper and lower indices are often interchangable. This is why I brought up bases and metrics. Chapter 1 may have as well be titled tensors for the physical sciences: http://preposterousuniverse.com/grnotes/" Open the .dpf 1. Special Relativity and Flat Spacetime.

 

[Marin]

Thanks, Phrak!

I got it :)