Example of stochastic matrix of mapping

Primary tabs

\documentclass{article}
% this is the default PlanetMath preamble.  as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% need this for including graphics (\includegraphics)
\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}

% making logically defined graphics
%\usepackage{xypic}
% used for TeXing text within eps files
%\usepackage{psfrag}

% there are many more packages, add them here as you need them

% define commands here

\newcommand{\vecify}{{\mathcal V}}
\newcommand{\Act}{{A}}
\newcommand{\act}{{a}}
\newcommand{\Sit}{{S}}
\newcommand{\occ}{{v}}
\newcommand{\univ}{{\mathbf D}}
\newcommand{\uout}{{d_{out}}}
\newcommand{\uin}{{d_{in}}}
\newcommand{\mangle}{{\mathbf C}}

\newcommand{\psheaf}{{\mathcal F}}
\newcommand{\scat}{{\mathtt{Stoch}}}
\newcommand{\subs}{{\mathtt{Sys}}}
\newcommand{\mcat}{{\mathtt{Meas}}}
\newcommand{\eop}{{$\blacksquare$}}
\newcommand{\eod}{{${}$\\}}
\newcommand{\bra}{{\langle}}
\newcommand{\ket}{{\rangle}}

\newcommand{\cN}{{\mathcal N}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\fm}{{\mathfrak m}}
\newcommand{\cP}{{\mathcal P}}

\begin{document}
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:

\begin{align*}
f(a) &= d \\
f(b) &= d \\
f(c) &= e
\end{align*}

Then $\vecify X$ 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 $\vecify Y$ 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
${\vecify f} = \begin{pmatrix} 1 & 1 & 0 \cr 0 & 0 & 1 \end{pmatrix} .$

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:

\begin{align*}
f(d) &=& a \\
f(e) &=& b
\end{align*}

Following the same procedures as above, for this map we find that
${\vecify i} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$
and
$({\vecify i})^\natural = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$

nd{document}