Added to Favorites

Related Searches

Nearby Words

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}.

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)

This article is licensed under the GNU Free Documentation License.

Last updated on Sunday September 21, 2008 at 18:38:51 PDT (GMT -0700)

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

This article is licensed under the GNU Free Documentation License.

Last updated on Sunday September 21, 2008 at 18:38:51 PDT (GMT -0700)

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

Copyright © 2015 Dictionary.com, LLC. All rights reserved.