Matrix addition
Template:Use American English Template:Short description In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
For a 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>
DefinitionEdit
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 Template:Nowrap, is computed by adding corresponding elements of A and B:Template:SfnTemplate:Sfn
- <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 Template:Nowrap):<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 Template:Nowrap, 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>