Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.
The matrix direct sum is defined by
 |  |  |
(1)
|
 |  | ![[A_1 ; A_2 ; ... ; A_n]](/images/equations/DirectSum/Inline6.svg) |
(2)
|
(Ayres 1962, pp. 13-14).
The direct sum of two subspaces 
and 
is the sum of subspaces in which
and 
have only the zero vector in
common (Rosen 2000, p. 357).
The significant property of the direct sum is that it is the coproduct in the category of modules
(i.e., a module direct sum). This general definition
gives as a consequence the definition of the direct sum 
of Abelian groups 
and 
(since they are 
-modules, i.e., modules
over the integers) and the direct sum of vector
spaces (since they are modules over a field).
Note that the direct sum of a finite family
of Abelian groups is the same as the group
direct product. The term direct sum is most often used for Abelian
groups, but is sometimes also used for non-Abelian
groups.
Note that direct products and direct sums differ for infinite index sets. An element of the direct sum is zero for all but a finite number of entries, while an element of the direct product can have all nonzero entries.
