% m211oh98.tex (prev.: *oh94.*) -  Primitive Stochastic Matrices
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\begin{center}\tcr{\rmH Primitive Stochastic Matrices} \end{center} 
\huge

\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
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\begin{description}
\item[\tcr{Definition:}]
A \tcg{square stochastic} matrix \mc{A} is called \tcr{primitive} if for \tcg{some} \mc{n \ge 1} the matrix \mc{A^n} has \tcg{no} entries equal to \mc{0}.

\item[\tcr{Perron's Theorem:}] \tcg{(O. Perron, 1907)} If \mc{A} is a \tcg{primitive stochastic} matrix, then \mc{\lm_{\scr1} = 1} is a \tcr{simple, dominant} eigenvalue of \mc{A} and the associated eigenspace
is generated by a \tcr{unique} \tcg{stochastic} vector \mc{\vec{p}} called the \tcr{Perron vector} of \mc{A}; i.e. 
$$\tcb{
E_A(1) \;=\; \{c\vec{p}\}.}
$$
Furthermore, \mc{\vec{p}} has the property that \tcg{none} of its entries is equal to \mc{0}.

\item[\tcr{Corollary:}] If \mc{A} is \tcg{primitive} and \tcg{stochastic}, then \mc{A} is \tcr{power convergent} with limit
$$
\tcb{\lim_{\scr n\ra\infty} A^{\scr n} \;=\; \underbrace{(\vec{p}| \vec{p}| \ldots | \vec{p})}_{\scr m},}
$$
where \mc{\vec{p}} denotes the \tcg{Perron vector} of \mc{A}.
Moreover, for any vector \mc{\vv{}} we have 
$$
\tcb{\lim_{\scr n\ra\infty} A^{\scr n}\vv{} \;=\; c\vec{p},\quad \mbox{where}\; c = \Sigma_{\scr m} \vec{v}.}
$$
\end{description} 
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