I am new to Tensor algebra and still getting used to many of the terms. I have the below matrix equation and I wish to write it in Ricci calculus notation but am struggling:
$$(A \otimes_k B)(C \otimes_k D)$$
Where $\otimes_k$ is the Kronecker product. I understand that for a matrix, say $A$, you can express it in terms of it's (scalar) components, $A_j^i$ and a product of a basis vector, $\vec{e}_i$, and a basis covector $\epsilon^j$ as:
$$A_j^i(\vec{e}_i \otimes \epsilon^j)$$
Where $\otimes$ is the Tensor product. And I believe that the Kronecker product of two matrices, $A$ and $B$, would look something like this:
$$A_j^iB^m_n(\vec{e}_i \otimes \epsilon^j \otimes \vec{e}_n \otimes \epsilon^m)$$
However, I am unsure what happens with the "normal" matrix product in my equation above. I know that is $A$, $B$, $C$, and $D$ have the appropriate dimensions I can apply the mixed product property of the Kronecker product to get:
$$AC \otimes_k BD$$
Which I imagine would be something like:
$$A_k^iC^k_jB^n_qD^q_m(\vec{e}_i \otimes \epsilon^j \otimes \vec{e}_n \otimes \epsilon^m)$$
However, I want to write the expression in Tensor notation without the assumption that $A$ and $C$ share a dimension, and likewise with $B$ and $D$. How can I do this? Is there a convention? Where can I learn more about converting expressions such as this one?
Best Answer
One correct way to describe the Kronecker product is $$ A \otimes_k B = A^i_j B^m_n (e_i \otimes_k e_m)\otimes (e^j \otimes_k e^n). $$ With that, you should be able to prove the mixed product property by simplifying $$ (A \otimes_k B)(C \otimes_k D) = A^i_j B^m_n C^j_k B^n_p [(e_i \otimes_k e_m)\otimes (e^j \otimes_k e^n)] [(e_j \otimes_k e_n)\otimes (e^k \otimes_k e^p)], $$ using the fact that $$ (e^i \otimes_k e^p)(e_j \otimes_k e_q) = \delta_{ij}\delta_{pq}. $$