A tensor is a mathematical/geometrical object which has certain transformation properties under change of coordinates.
http://en.wikipedia.org/wiki/Tensor
You can have a tensor of rank n covariant and rank m contravariant
and a tensor of rank l covariant and rank k contravariant
Then their product will be a rank (n+l) covariant and (m+k) contravariant.
T^{i_{1}i_{2}...i_{m}}_{j_{1}j_{2}...j_{n}} T^{r_{1}r_{2}...r_{k}}_{w_{1}w_{2}...w_{l}}= R^{i_{1}i_{2}...i_{m}r_{1}r_{2}...r_{k}}_{j_{1}j_{2}...j_{n}w_{1}w_{2}...w_{l}}
If you are talking about the tensor product for example between two matrices, the idea is almost the same...
In that case you define:
for A \in K^{p \times q} and B \in K^{r \times s}
A \times B \in K^{pr \times qs}
So a matrix let's say 3x2 multiplied by tensor product with another 4x5 will give as a result a matrix 12x10.
in element notation, the product is defined as:
( A \times B)_{(ik)(jl)}= A_{ij}B_{kl}
or in matrix form you write for EACH element the matrix B and multiply it in the first line with A[11], A[12],...,A[1q]
the second line A[21],...A[2q] etc... (see attachment)So if what you ask for are the pauli matrices \sigma^{1,2} then the result will be a 4x4 matrix, with upper left 2x2 block the \sigma_{11}^{1} \sigma^{2}, the upper right 2x2 block \sigma_{12}^{1} \sigma^{2}, the lower left 2x2 block \sigma_{21}^{1} \sigma^{2}, and the lower right 2x2 block \sigma_{22}^{1} \sigma^{2}.
In this case though \sigma^{1} has zero diagonal elements, so the upper left and lower right 2x2 blocks are zero, the other off diagonal blocks are the \sigma^{2} matrices. If i did it correctly...
