Define $A \square B = A \otimes I + I \otimes B$.
Is there a name for the operation $\square$, when considered as a type of matrix product? It's a generalisation of the Cartesian product for graphs (also denoted $\square$), but I'm more interested in what can be said about it from a linear algebra point of view. Searching for "matrix Cartesian product" didn't turn up anything.
Clearly this is quite nicely behaved in terms of eigendecomposition, since if $\mu$ and $\nu$ are eigenvalues of $A$ and $B$ then $\mu + \nu$ is an eigenvalue of $A\square B$, and the corresponding eigenvectors are also related in a simple way. Is there more that can be said about its properties than this?
In addition, I am also interested in matrices that have the same pattern of zero and non-zero entries as a matrix of the form $A\square B$, but which cannot be factored into that form. Is there anything useful that can be said about the spectrum of such matrices? (If necessary, assume that the non-zero elements are all positive.)
Edit: I know the answer now. (See my own answer below, which I can't accept yet for some reason.) I've asked the part about matrices with the same sign pattern as a separate question.