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
Canonical name	ExampleOfStochasticMatrixOfMapping
Date of creation	2014-04-28 3:33:09
Last modified on	2014-04-28 3:33:09
Owner	rspuzio (6075)
Last modified by	PMBookProject (1000683)
Numerical id	21
Author	rspuzio (1000683)
Entry type	Example