Definition:Matrix Addition
Definition
Let $\struct {R, +, \cdot}$ be a ring.
Matrix Entrywise Addition
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 $\mathbf B$.
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:
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.