Block Matrix - Direct Sum

For any arbitrary matrices A (of size m × n) and B (of size p × q), we have the direct sum of A and B, denoted by A B and defined as

\mathbf{A} \oplus \mathbf{B} = \begin{bmatrix} a_{11} & \cdots & a_{1n} & 0 & \cdots & 0 \\ \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\ a_{m 1} & \cdots & a_{mn} & 0 & \cdots & 0 \\ 0 & \cdots & 0 & b_{11} & \cdots & b_{1q} \\ \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\ 0 & \cdots & 0 & b_{p1} & \cdots & b_{pq} \end{bmatrix}.

For instance,

\begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 1 \end{bmatrix} \oplus \begin{bmatrix} 1 & 6 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 2 & 0 & 0 \\ 2 & 3 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}.

This operation generalizes naturally to arbitrary dimensioned arrays (provided that A and B have the same number of dimensions).

Note that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices.

Read more about this topic:  Block Matrix

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