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 ⊕_(i = 1)^n A_i | = | diag(A_1, A_2, ..., A_n) | = | [A_1 | | | | A_2 | | | | ⋱ | | | | A_n] (Ayres 1962, pp. 13-14). The direct sum of two subspaces U and W is the sum of subspaces in which U and W have only the zero vector in common.
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