Redirected from Direct sum (Matrix)
\begin{bmatrix} 1 & 3 \\ 1 & 0 \\ 1 & 2 \end{bmatrix}+
\begin{bmatrix} 0 & 0 \\ 7 & 5 \\ 2 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 3 \\ 8 & 5 \\ 3 & 3 \end{bmatrix}</math>
The m × n matrices with matrix addition as operation form an abelian group.
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
A \oplus 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}</math>
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}</math>
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