Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Kronecker sum.

Two matrices must have an equal number of rows and columns to be added.[1] In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:[2][3]

Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted AB, is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example:

Another operation, which is used less often, is the direct sum (denoted by ⊕). Note the Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size m × n and B of size p × q is a matrix of size (m + p) × (n + q) defined as:[6][2]

The direct sum of matrices is a special type of block matrix. In particular, the direct sum of square matrices is a block diagonal matrix.

The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices.