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Matrix addition : Direct sum (Matrix)

The usual matrix addition is defined for two 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 AB 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>

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