Definition:Matrix Direct Sum

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:


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

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 $$\left({m + p}\right) \times \left({n + q}\right)$$.