Encyclopedia > Direct sum (Matrix)

  Article Content

Matrix addition

Redirected from 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>



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Explorer

... explorer, environmentalist Nathaniel Pryor[?], (c 1785-1850), US explorer Pytheas, (died 1984), Greek explorer who visited Britain and other north and northwest ...

 
 
 
This page was created in 24.6 ms