Matrix addition

Jump to: navigation, search

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.

1 Entrywise sum
2 Direct sum
3 See also
4 External links

[ 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. 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

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

For instance,

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

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.

In general, thee direct sum of n matrices is:

$\bigoplus_{i=1}^{n} A_{i} = \mbox{diag}( A_1, A_2, A_3, \ldots, A_n)= \begin{bmatrix} \begin{matrix} A_1 & \\ & A_2 \end{matrix} & 0 \\ 0 & \begin{matrix} \ddots & \\ & A_n \end{matrix} \end{bmatrix}.$