Matrix addition
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
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  and defined as
For instance,
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