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In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
Entrywise sum
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. For example
Direct sum
Another operation, which is used less often, is the direct sum. We can form the direct sum of any pair of matrices A and B. say of size m × n and p × q, respectively. The direct sum is a matrix of size (m + p) × (n + q) matrix 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.
In general, we can write the direct sum of n matrices as: