Definition:Kronecker Sum

Definition
Let $\mathbf A = \left[{a}\right]_n$ and $\mathbf B = \left[{b}\right]_m$ be square matrices with dimensions $n$ and $m$ respectively.

The Kronecker sum of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \oplus \mathbf B$ and is defined as:


 * $\mathbf A \oplus \mathbf B = \left({\mathbf A \otimes \mathbf I_m}\right) + \left({\mathbf I_n \otimes \mathbf B}\right)$

where:
 * $\otimes$ denotes the Kronecker product
 * $+$ denotes conventional matrix entrywise addition
 * $\mathbf I_m$ and $\mathbf I_n$ are the identity matrices of order $m$ and $n$ respectively.

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

Caution
Do not confuse this operation with the matrix direct sum, which is a completely different operation (although using the same notation).