Editing Block matrix (section)

====Direct sum====
{{See also|Matrix addition#Direct sum}}
For any arbitrary matrices '''A''' (of size ''m''&nbsp;×&nbsp;''n'') and '''B''' (of size ''p''&nbsp;×&nbsp;''q''), we have the '''direct sum''' of '''A''' and '''B''', denoted by '''A'''&nbsp;<math>\oplus</math>&nbsp;'''B''' and defined as 
 
:<math>
  {A} \oplus {B} =
  \begin{bmatrix}
    a_{11} & \cdots & a_{1n} &      0 & \cdots &      0 \\
    \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
    a_{m1} & \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><ref name=":1" />

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>

This operation generalizes naturally to arbitrary dimensioned arrays (provided that '''A''' and '''B''' have the same number of dimensions).

Note that any element in the [[direct sum of vector spaces|direct sum]] of two [[vector space]]s of matrices could be represented as a direct sum of two matrices.