\begin{definition}[Definition:Kronecker Sum]
Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_m$ be square matrices with orders $n$ and $m$ respectively.
The '''Kronecker sum''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \oplus \mathbf B$ and is defined as:
:$\mathbf A \oplus \mathbf B = \paren {\mathbf A \otimes \mathbf I_m} + \paren {\mathbf I_n \otimes \mathbf B}$
where:
:$\otimes$ denotes the Kronecker product
:$+$ denotes conventional matrix entrywise addition
:$\mathbf I_m$ and $\mathbf I_n$ are the unit matrices of order $m$ and $n$ respectively.
{{refactor|Raise the following statement as a page in its own right.}}
From the above, it follows that $\mathbf A \oplus \mathbf B$ is a square matrix with order $m n$.
\end{definition}