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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,

Note that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices.

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