Editing Matrix addition (section)

==<span id="directsum"></span>Direct sum==
Another operation, which is used less often, is the direct sum (denoted by ⊕). The Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices '''A''' of size ''m'' &times; ''n'' and '''B''' of size ''p'' &times; ''q'' is a matrix of size (''m'' + ''p'') &times; (''n'' + ''q'') defined as:<ref>{{MathWorld |id=MatrixDirectSum |title=Matrix Direct Sum}}</ref>{{sfn|Lipschutz|Lipson|2017}}

:'''<math>
  \mathbf{A} \oplus \mathbf{B} =
  \begin{bmatrix} \mathbf{A} & \boldsymbol{0} \\ \boldsymbol{0} & \mathbf{B} \end{bmatrix} =
  \begin{bmatrix}
     a_{11} & \cdots & a_{1n} &      0 & \cdots &      0 \\
     \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
    a_{m 1} & \cdots & a_{mn} &      0 & \cdots &      0 \\
          0 & \cdots &      0 & b_{11} & \cdots &  b_{1q} \\
     \vdots & \ddots & \vdots & \vdots & \ddots & \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>

The direct sum of matrices is a special type of [[block matrix]]. In particular, the direct sum of square matrices is a [[Block matrix#Block diagonal matrices|block diagonal matrix]].

The [[adjacency matrix]] of the union of disjoint [[Graph (discrete mathematics)|graphs]] (or [[multigraph]]s) is the direct sum of their adjacency matrices. Any element in the [[Direct sum of modules|direct sum]] of two [[vector space]]s of matrices can be represented as a direct sum of two matrices.

In general, the direct sum of ''n'' matrices is:{{sfn|Lipschutz|Lipson|2017}}
:<math>
\bigoplus_{i=1}^{n} \mathbf{A}_{i} = \operatorname{diag}( \mathbf{A}_1, \mathbf{A}_2, \mathbf{A}_3, \ldots, \mathbf{A}_n) =
\begin{bmatrix}
  \mathbf{A}_1 & \boldsymbol{0} & \cdots & \boldsymbol{0} \\
  \boldsymbol{0} & \mathbf{A}_2 & \cdots & \boldsymbol{0} \\
  \vdots & \vdots & \ddots & \vdots \\
  \boldsymbol{0} & \boldsymbol{0} & \cdots & \mathbf{A}_n \\
\end{bmatrix}\,\!</math>

where the zeros are actually blocks of zeros (i.e., zero matrices).