Definitions

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there is another operation which could also be considered as a kind of addition for matrices.

## Entrywise sum

The usual matrix addition is defined for two matrices of the 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:


begin{bmatrix}
1 & 3
1 & 0
1 & 2
end{bmatrix} + begin{bmatrix}
0 & 0
7 & 5
2 & 1
end{bmatrix} = begin{bmatrix}
1+0 & 3+0
1+7 & 0+5
1+2 & 2+1
end{bmatrix} = begin{bmatrix}
1 & 3
8 & 5
3 & 3
end{bmatrix}

We can also subtract one matrix from another, as long as they have the same dimensions. A - B is computed by subtracting corresponding elements of A and B, and has the same dimensions as A and B. For example:


begin{bmatrix}
1 & 3
1 & 0     1 & 2
end{bmatrix} - begin{bmatrix}
0 & 0
7 & 5
2 & 1
end{bmatrix} = begin{bmatrix}
1-0 & 3-0
1-7 & 0-5
1-2 & 2-1
end{bmatrix} = begin{bmatrix}
1 & 3
-6 & -5
-1 & 1
end{bmatrix}

## 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


A oplus B =
begin{bmatrix} A & 0 0 & B end{bmatrix} = begin{bmatrix} a_{11} & cdots & a_{1n} & 0 & cdots & 0
vdots & cdots & vdots & vdots & cdots & vdots
a_{m 1} & cdots & a_{mn} & 0 & cdots & 0 0 & cdots & 0 & b_{11} & cdots & b_{1q}
vdots & cdots & vdots & vdots & cdots & vdots
0 & cdots & 0 & b_{p1} & cdots & b_{pq} end{bmatrix}

For instance,


begin{bmatrix}
1 & 3 & 2
2 & 3 & 1
end{bmatrix} oplus begin{bmatrix}
1 & 6
0 & 1
end{bmatrix} = begin{bmatrix}
1 & 3 & 2 & 0 & 0
2 & 3 & 1 & 0 & 0
0 & 0 & 0 & 1 & 6
0 & 0 & 0 & 0 & 1
end{bmatrix}

Note that the direct sum of two square matrices could represent the adjacency matrix of a graph or multigraph with one component for each direct addend.

Note also 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:


bigoplus_{i=1}^{n} A_{i} = mbox{diag}(A_1, A_2, A_3, ldots, A_n)= begin{bmatrix}
A_1  &  &  &
& A_2  &   &
&   & ddots  &
&   &   & A_n
end{bmatrix}.