So I'm having trouble to compute tensor contractions with "actual" numbers from the matrix representations of the tensors. I have only seen abstract theoretical examples on the internet so I'm asking for a bit of help on how to find the contractions given the expressions of the tensors and the pair of indices where we will carry out the contraction. I'll show a simple example and I hope you can help me.
Let's suppose we have two tensors of the type (1,1) (that means 1 contravariant, 1 covariant). They will be called Y and Z and knowing their coordinate forms, we can represent them through matrices in this way:
Y = \begin{pmatrix}1&-1\\2&3\end{pmatrix}
Z = \begin{pmatrix}-1&0\\1&2\end{pmatrix}
Now, we could compute the Kronecker product Y x Z in order to get a type (2,2) tensor (2 contravariant, 2 covariant). The result would be:
\begin{pmatrix}-1&0&1&0\\1&2&-1&-2\\-2&0&-3&0\\2&4&3&6\end{pmatrix}
So, how could we get the contractions of this (2,2) tensor (with actual numbers instead of the parameters that you see in other examples) ?.
Note that here we have 2 contravariant indices and 2 covariant indices so there are 4 possible contractions depending on the pair of indices that you choose, I guess.
As far as I know, all possible contractions with the selection of pairs of indices are the following:
– 1st covariant index and 1st contravariant index.
– 1st covariant index and 2nd contravariant index.
– 2nd covariant index and 1st contravariant index.
– 2nd covariant index and 2nd contravariant index.
So what matrices would be the result of those contractions. Just in case I didn't state it clearly before, these matrices are the representation of tensor of which we know the coefficients of their coordinate forms (explicit forms in a specific basis, and that info is known). So we want to perform contractions in the last 4×4 matrix that I showed (which represents a (2,2) tensor) and there a 4 different possibilities. My question has to do with the fact that I can't find a way to do this with actual numbers and I can't figure out the result for each possible contraction (I also don't know what the differences would be in the calculation of a contraction with a certain pair of indices or another).
I would really appreciate that someone could find a specific result for the contractions that I proposed (I guess the calculation is easy but I just don't know how it could be done). Thank you really much for reading.
Best Answer
Representing tensors using matrix notation is often confusing, but let's assume that
$Y = \begin{pmatrix}y^1_1&y^1_2\\v^2_1&y^2_2\end{pmatrix}$
and similarly for Z. If $W = Y \times Z$ then the components of $W$ are
$w^{ik}_{jl} = y^i_j z^k_l$
You have represented W as a 4x4 matrix, but it would be more accurate to represent it as a 2x2 matrix, each of whose entries is another 2x2 matrix.
Anyway, the four possible contractions of W are:
(1): $w^{ik}_{il} = y^i_i z^k_l$
(2): $w^{ik}_{jk} = y^i_j z^k_k$
(3): $w^{ik}_{ji} = y^i_j z^k_i$
(4): $w^{ij}_{jl} = y^i_j z^j_l$
In terms of matrix operations:
(1) is the component representation of $Tr(Y)Z$
(2) is the component representation of $Tr(Z)Y$
(3) is the component representation of $YZ$
(4) is the component representation of $ZY$