Editing Matrix addition

[[File:Matrix addition qtl2.svg|thumb|Illustration of the addition of two matrices.]]
{{Use American English|date = January 2019}}
{{Short description|Notions of sums for matrices in linear algebra}}
In [[mathematics]], '''matrix addition''' is the operation of adding two [[matrix (mathematics)|matrices]] by adding the corresponding entries together. 

For a [[Vector space|vector]], <math>\vec{v}\!</math>, adding two matrices would have the geometric effect of applying each matrix transformation separately onto <math>\vec{v}\!</math>, then adding the transformed vectors.

:<math>\mathbf{A}\vec{v} + \mathbf{B}\vec{v} = (\mathbf{A} + \mathbf{B})\vec{v}\!</math>

==Definition==
Two matrices must have an equal number of rows and columns to be added.<ref>Elementary Linear Algebra by Rorres Anton 10e p53</ref> In which case, the sum of two matrices '''A''' and '''B''' will be a matrix which has the same number of rows and columns as '''A''' and '''B'''. The sum of '''A''' and '''B''', denoted {{nowrap|'''A''' + '''B'''}}, is computed by adding corresponding elements of '''A''' and '''B''':{{sfn|Lipschutz|Lipson|2017}}{{sfn|Riley|Hobson|Bence|2006}}

:<math>\begin{align}
\mathbf{A}+\mathbf{B} & = \begin{bmatrix}
 a_{11} & a_{12} & \cdots & a_{1n} \\
 a_{21} & a_{22} & \cdots & a_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +

\begin{bmatrix}
 b_{11} & b_{12} & \cdots & b_{1n} \\
 b_{21} & b_{22} & \cdots & b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix} \\
& = \begin{bmatrix}
 a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
 a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\

\end{align}\,\!</math>
Or more concisely (assuming that {{nowrap|1='''A''' + '''B''' = '''C'''}}):<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Matrix Addition|url=https://mathworld.wolfram.com/MatrixAddition.html|access-date=2020-09-07|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|title=Finding the Sum and Difference of Two Matrices {{!}} College Algebra|url=https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/finding-the-sum-and-difference-of-two-matrices/|access-date=2020-09-07|website=courses.lumenlearning.com}}</ref>
:<math>c_{ij}=a_{ij}+b_{ij}</math>

For example:

:<math>
  \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}
</math>

Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of '''A''' and '''B''', denoted {{nowrap|'''A''' &minus; '''B'''}}, is computed by subtracting elements of '''B''' from corresponding elements of '''A''', and has the same dimensions as '''A''' and '''B'''. For example:

:<math>
\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}
</math>

==See also==
* [[Matrix multiplication]]
* [[Vector addition]]
* [[Direct sum of matrices]]
* [[Kronecker sum]]

== Notes ==
{{reflist|2}}

==References==
* {{cite book | last1=Lipschutz | first1=Seymour | last2=Lipson | first2=Marc | title=Schaum's Outline of Linear Algebra | edition=6 | publisher=McGraw-Hill Education | year=2017 | isbn=9781260011449}}
* {{cite book | last1=Riley | first1=K.F. | last2=Hobson | first2=M.P. |last3=Bence |first3=S.J. | title=Mathematical methods for physics and engineering | edition=3 | publisher=Cambridge University Press | year=2006 | doi=10.1017/CBO9780511810763 | isbn=978-0-521-86153-3 | url=https://archive.org/details/mathematicalmeth00rile |url-access=registration}}

==External links==
* [https://web.archive.org/web/20120514184901/http://www.aps.uoguelph.ca/~lrs/ABMethods/NOTES/CDmatrix.pdf Matrix Algebra and R]

[[Category:Linear algebra]]
[[Category:Bilinear maps]]