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>

However, there are other operations that could also be considered [[addition]] for matrices, such as the [[direct sum]] and the [[Kronecker sum]].

==Entrywise sum==
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>

==<span id="directsum"></span>Direct sum==
Another operation, which is used less often, is the direct sum (denoted by ⊕). The Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices '''A''' of size ''m'' &times; ''n'' and '''B''' of size ''p'' &times; ''q'' is a matrix of size (''m'' + ''p'') &times; (''n'' + ''q'') defined as:<ref>{{MathWorld |id=MatrixDirectSum |title=Matrix Direct Sum}}</ref>{{sfn|Lipschutz|Lipson|2017}}

:'''<math>
  \mathbf{A} \oplus \mathbf{B} =
  \begin{bmatrix} \mathbf{A} & \boldsymbol{0} \\ \boldsymbol{0} & \mathbf{B} \end{bmatrix} =
  \begin{bmatrix}
     a_{11} & \cdots & a_{1n} &      0 & \cdots &      0 \\
     \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
    a_{m 1} & \cdots & a_{mn} &      0 & \cdots &      0 \\
          0 & \cdots &      0 & b_{11} & \cdots &  b_{1q} \\
     \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
          0 & \cdots &      0 & b_{p1} & \cdots &  b_{pq}
  \end{bmatrix}
</math>'''

For instance,

:<math>
  \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}
</math>

The direct sum of matrices is a special type of [[block matrix]]. In particular, the direct sum of square matrices is a [[Block matrix#Block diagonal matrices|block diagonal matrix]].

The [[adjacency matrix]] of the union of disjoint [[Graph (discrete mathematics)|graphs]] (or [[multigraph]]s) is the direct sum of their adjacency matrices. Any element in the [[Direct sum of modules|direct sum]] of two [[vector space]]s of matrices can be represented as a direct sum of two matrices.

In general, the direct sum of ''n'' matrices is:{{sfn|Lipschutz|Lipson|2017}}
:<math>
\bigoplus_{i=1}^{n} \mathbf{A}_{i} = \operatorname{diag}( \mathbf{A}_1, \mathbf{A}_2, \mathbf{A}_3, \ldots, \mathbf{A}_n) =
\begin{bmatrix}
  \mathbf{A}_1 & \boldsymbol{0} & \cdots & \boldsymbol{0} \\
  \boldsymbol{0} & \mathbf{A}_2 & \cdots & \boldsymbol{0} \\
  \vdots & \vdots & \ddots & \vdots \\
  \boldsymbol{0} & \boldsymbol{0} & \cdots & \mathbf{A}_n \\
\end{bmatrix}\,\!</math>

where the zeros are actually blocks of zeros (i.e., zero matrices).

==Kronecker sum==
{{main article|Kronecker sum}}
The Kronecker sum is different from the direct sum, but is also denoted by ⊕. It is defined using the [[Kronecker product]] ⊗ and normal matrix addition. If '''A''' is ''n''-by-''n'', '''B''' is ''m''-by-''m'' and   <math>\mathbf{I}_k</math> denotes the ''k''-by-''k'' [[identity matrix]] then the Kronecker sum is defined by:
:<math> \mathbf{A} \oplus \mathbf{B} = \mathbf{A} \otimes \mathbf{I}_m + \mathbf{I}_n \otimes \mathbf{B}. </math>

==See also==
* [[Matrix multiplication]]
* [[Vector addition]]

== 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==
*{{PlanetMath |urlname=DirectSumOfMatrices |title= Direct sum of matrices}}
* [https://web.archive.org/web/20120426083541/http://drexel28.wordpress.com/2010/12/22/direct-sum-of-linear-transformations-and-direct-sum-of-matrices-pt-iii/ Abstract nonsense: Direct Sum of Linear Transformations and Direct Sum of Matrices]
* [http://www.mymathlib.com/matrices/arithmetic/direct_sum.html Mathematics Source Library: Arithmetic Matrix Operations]
* [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]]