Derive the Matrix Representation of the Tensor Product

Question

I'm stuck on the following problem regarding the relationship between the Tensor Product and the Kronecker product: PROBLEM

How does one go about showing that the matrix representation of the tensor product is the Kronecker Product? (I believe that's what the question is asking).

I'd appreciate any insights on this problem!

Answer 1

You asked for insights, but you've already had one:

You're right. That is what the question is asking.

You might want to ask "if I compute $s = (A \otimes B)(v_1 \otimes x_1)$, how can I write $s$ as a linear combination of the vectors $w_i \otimes y_j$?" As a first step, you might want to write $Av_1$ as a linear combination of the $w_i$s.