Matrix addition#Kronecker sum
{{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 matrices by adding the corresponding entries together.
For a vector, , adding two matrices would have the geometric effect of applying each matrix transformation separately onto , then adding the transformed vectors.
:
Definition
Two matrices must have an equal number of rows and columns to be added.Elementary Linear Algebra by Rorres Anton 10e p53 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}}
:
\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}\,\!
Or more concisely (assuming that {{nowrap|1=A + B = C}}):{{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}}{{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}}
:
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}
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 − B}}, is computed by subtracting elements of B from corresponding elements of A, 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}
See also
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]