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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:## Direct sum

## See also

## External links

- $$

1 & 3

1 & 0

1 & 2end{bmatrix} + begin{bmatrix}

0 & 0

7 & 5

2 & 1end{bmatrix} = begin{bmatrix}

1+0 & 3+0

1+7 & 0+5

1+2 & 2+1end{bmatrix} = begin{bmatrix}

1 & 3

8 & 5

3 & 3end{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:

- $$

1 & 3

1 & 0 1 & 2end{bmatrix} - begin{bmatrix}

0 & 0

7 & 5

2 & 1end{bmatrix} = begin{bmatrix}

1-0 & 3-0

1-7 & 0-5

1-2 & 2-1end{bmatrix} = begin{bmatrix}

1 & 3

-6 & -5

-1 & 1end{bmatrix}

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 & vdotsa_{m 1} & cdots & a_{mn} & 0 & cdots & 0 0 & cdots & 0 & b_{11} & cdots & b_{1q}

vdots & cdots & vdots & vdots & cdots & vdots0 & cdots & 0 & b_{p1} & cdots & b_{pq} end{bmatrix}

For instance,

- $$

1 & 3 & 2

2 & 3 & 1end{bmatrix} oplus begin{bmatrix}

1 & 6

0 & 1end{bmatrix} = begin{bmatrix}

1 & 3 & 2 & 0 & 0

2 & 3 & 1 & 0 & 0

0 & 0 & 0 & 1 & 6

0 & 0 & 0 & 0 & 1end{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:

- $$

A_1 & & &

& A_2 & &

& & ddots &

& & & A_nend{bmatrix}.

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Last updated on Sunday September 21, 2008 at 17:38:51 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday September 21, 2008 at 17:38:51 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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