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.

  • $\begingroup$ Downvoter care to explain?? $\endgroup$
    – N. Virgo
    Apr 2, 2016 at 1:37

1 Answer 1


Ah, I found the answer already. It is called the Kronecker sum, and usually denoted $\oplus$ rather than $\square$. I had been thrown off by my assumption that it would be considered a product - but calling it a sum makes sense, since it has the extremely nice property that $$ \exp(A \oplus B) = \exp(A)\otimes \exp(B), $$ where $\exp$ is the matrix exponential. This explains the eigenvalue property that I had noted.

Of course, this doesn't answer my question about matrices that share the same sign pattern. However, I will post that as a separate question now that I know the correct name for the Kronecker sum.


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