How can we simplify tensor expressions in Mathematica 9 using the mixed-product identity $(A\otimes B)(C \otimes D) \equiv AC \otimes BD$ ?
Is it possible to implement this kind of evaluations using the new Mathematica 9 tensor capabilities?
The following expression is false:
TensorProduct[a, b].TensorProduct[c, d] === TensorProduct[ a.c, b.d]
I don't really need to prove this identity, but rather to use it for simplifying some expressions. In my case, $a$ and $c$ are some (unknown) symbolic matrices, while $b$ and $d$ are explicit integer $2\times 2$ matrices. I'd like Mathematica to evaluate the matrix product between $b$ and $d$ explicitly.
For instance, when
as the product of two Pauli matrices gives another Pauli matrix, I'd like to obtain the simplified result
TensorProduct[a.c, -I PauliMatrix]