# Definition:Kronecker Sum

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## 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.

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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**:

## Source of Name

This entry was named for Leopold Kronecker.