% m211oh97.tex (prev.: *oh93.*) - Stochastic Matrices and Markov Chains
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\begin{center}\tcr{\rmH Stochastic Matrices\vspace{10pt}\\ and Markov Chains} \end{center} 
\huge

\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
\newcommand{\vv}[1]{\vec{v}_{\scr #1}}
\newcommand{\rank}{\mbox{rank}}
\newcommand{\mult}{\mbox{mult}}


\begin{description}
\item[\tcg{Definition:}] A real \mc{p\times q} matrix \mc{A = (a_{\scr ij})} is called \tcr{stochastic} if\vspace{-10pt}
\tcb{\begin{eqnarray}
& &a_{\scr ij}  \ge  0, \; \mbox{for all} \; i, j; \\
& &\Sigma_p\cdot A  =  \Sigma_q, \quad \label{2}
\end{eqnarray}}
where \mc{\Sigma_p = \underbrace{(1, 1, \ldots, 1)}_{\scr p} \in \R^{\scr p}}.  

A (homogeneous) \tcr{Markov chain} is a \tcg{discrete linear system}\vspace{-10pt} 
\tcb{$$
\vv{n+1} = A \vv{n}
$$}
in which \begin{minipage}[t]{13cm} \mc{\vv{n}} is a \tcg{stochastic vector}, and \\
\mc{A} is a \tcg{stochastic matrix}.
\end{minipage}\vspace{8pt}

\item[\tcr{Remarks:}] \tcr{1)} Condition (\ref{2}) means: \tcg{the sum of each column} of \mc{A} is equal to \mc{1}.

\tcr{2)} Markov chains were named after \tcr{A.A. Markov} \tcg{(1856 -- 1922)} by \tcr{A.N. Kolmogorov} in \tcg{1935}.\vspace{-5pt}

\item[\tcr{Properties:}] \tcr{1)} \mc{A, B} \tcg{stochastic} \tcr{$\Rightarrow$} \mc{AB} \tcg{stochastic}.

\tcr{2)} If \mc{A} is a square stochastic matrix, then

a) \mc{\lm_{\scr1} = 1} is a \tcr{regular} \tcg{eigenvalue} of \mc{A};

b) \mc{|\lm_i| \le 1} for every eigenvalue \mc{\lm_i} of \mc{A}.  
\end{description} 
\end{titlepage}

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