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}.

Search another word or see matrix additionon Dictionary | Thesaurus |Spanish