Definition:Matrix Direct Sum

Definition
Let $\mathbf{A} = \left[{a}\right]_{m n}$ and $\mathbf{B} = \left[{b}\right]_{p q}$ 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 $\left({m + p}\right) \times \left({n + q}\right)$.