Direct sum (Matrix)
matrices of same dimensions.
The sum of two m-by-n matrices A and B, denoted by A + B, is again an m-by-n matrix computed by adding corresponding elements, i.e.,
(A + B)[i, j] = A[i, j] + B[i, j]. For example
- <math>
\begin{bmatrix}
1 & 3 \\
1 & 0 \\
1 & 2
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 \\
7 & 5 \\
2 & 1
\end{bmatrix}
=
\begin{bmatrix}
1+0 & 3+0 \\
1+7 & 0+5 \\
1+2 & 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 <math>A \oplus B<math> and defined as
- <math>
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,
- <math>
\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>
Note that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices.
