Definition:Kronecker Sum

Definition
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.

From the above, it follows that $\mathbf A \oplus \mathbf B$ is a square matrix with order $m n$.

Also see

 * Definition:Matrix Addition, where can be found different operations on matrices also referred to as addition:
 * Definition:Matrix Entrywise Addition
 * Definition:Matrix Direct Sum