Definition:Matrix Direct Sum

Definition
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 dimensions $m \times n$
 * $\mathbf B$ is a matrix with dimensions $p \times q$

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

Also see

 * Definition:Matrix Addition, where can be found different operations on matrices also referred to as addition:
 * Definition:Matrix Entrywise Addition
 * Definition:Kronecker Sum