# Example of stochastic matrix of mapping

In order to understand the notion of stochastic matrix associated to a mapping and its dual, we will work through a simple example. Let $X=\{a,b,c\}$ and let $Y=\{d,e\}$, and define the mapping $f\colon X\to Y$ as follows:

 $\displaystyle f(a)$ $\displaystyle=d$ $\displaystyle f(b)$ $\displaystyle=d$ $\displaystyle f(c)$ $\displaystyle=e$

Then ${\mathcal{V}}X$ is a 3-dimensional real vector space with basis

 $\delta_{a}=\begin{pmatrix}1\cr 0\cr 0\end{pmatrix},\qquad\delta_{b}=\begin{% pmatrix}0\cr 1\cr 0\end{pmatrix},\qquad\delta_{c}=\begin{pmatrix}0\cr 0\cr 1% \end{pmatrix}$

and ${\mathcal{V}}Y$ is a 3-dimensional real vector space with basis

 $\delta_{c}=\begin{pmatrix}1\cr 0\end{pmatrix},\qquad\delta_{d}=\begin{pmatrix}% 0\cr 1\end{pmatrix}$

and

 ${{\mathcal{V}}f}=\begin{pmatrix}1&1&0\cr 0&0&1\end{pmatrix}.$

To form the dual, we first renormalize the rows to sum to unity, then transpose:

 $\begin{pmatrix}1&1&0\cr 0&0&1\end{pmatrix}\xrightarrow{ren}\begin{pmatrix}% \frac{1}{2}&\frac{1}{2}&0\cr 0&0&1\end{pmatrix}\xrightarrow{*}\begin{pmatrix}% \frac{1}{2}&0\cr\frac{1}{2}&0\cr 0&1\end{pmatrix}$

Next, to illustrate inclusions, we shall examine the map $i\colon Y\hookrightarrow X$ defined as follows:

 $\displaystyle f(d)$ $\displaystyle=$ $\displaystyle a$ $\displaystyle f(e)$ $\displaystyle=$ $\displaystyle b$

Following the same procedures as above, for this map we find that

 ${{\mathcal{V}}i}=\begin{pmatrix}1&0\\ 0&1\\ 0&0\end{pmatrix}$

and

 $({{\mathcal{V}}i})^{\natural}=\begin{pmatrix}1&0&0\\ 0&1&0\end{pmatrix}$
Title Example of stochastic matrix of mapping ExampleOfStochasticMatrixOfMapping 2014-04-28 3:33:09 2014-04-28 3:33:09 rspuzio (6075) PMBookProject (1000683) 21 rspuzio (1000683) Example