Definition:Kronecker Sum

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 $$mn$$.

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