## Definition

Let $\struct {R, +, \cdot}$ be a ring.

This is the usual operation when matrix addition is specified without qualification):

Let $\mathbf A$ and $\mathbf B$ be matrices of numbers.

Let the orders of $\mathbf A$ and $\mathbf B$ both be $m \times n$.

Then the matrix entrywise sum of $\mathbf A$ and $\mathbf B$ is written $\mathbf A + \mathbf B$, and is defined as follows:

Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$.

Then:

$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} + b_{i j}$

Thus $\mathbf C = \sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding corresponding entries of $\mathbf A$ and $\mathbf B$.

That is, the matrix entrywise sum of $\mathbf A$ and $\mathbf B$ is the Hadamard product of $\mathbf A$ and $\mathbf B$ with respect to addition of numbers.

This operation is called matrix entrywise addition.

### Matrix Direct Sum

Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices.

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

$\mathbf A \oplus \mathbf B := \begin {bmatrix} \mathbf A & \mathbf 0 \\ \mathbf 0 & \mathbf B \end {bmatrix}$

where $\mathbf 0$ is a zero matrix, the upper-right $\mathbf 0$ being $m \times q$ and the lower left $\mathbf 0$ being $n \times p$.

Thus, if:

$\mathbf A$ is a matrix with order $m \times n$
$\mathbf B$ is a matrix with order $p \times q$

then $\mathbf A \oplus \mathbf B$ is a matrix with order $\paren {m + p} \times \paren {n + q}$.

### 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.