% m211oh93.tex (prev.: *oh89.*) - Power Convergent Matrices
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\begin{center}\tcr{\rmH Power Convergent Matrices} \end{center} 
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\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
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\newcommand{\rank}{\mbox{rank}}
\newcommand{\mult}{\mbox{mult}}

\begin{description}
\item[\tcr{Definition:}] 
An eigenvalue \mc{\lm_i} of \mc{A} is called \tcr{regular} if 
%\tcb{$$
\mc{\nu_{\scr A}(\lm) = m_{\scr A}(\lm),}
and is called \tcr{dominant} if \mc{|\lm_{\scr j}| <  |\lm_{\scr i}|}, for all eigenvalues \mc{\lm_{\scr j} \ne \lm_{\scr i}}. 

  
\item[\tcr{Theorem 6:}] An \mc{m \times m} matrix \mc{A} is \tcr{power convergent} \tcg{if and only if} \\
\tcr{either:} \mc{|\lm_{\scr i}| < 1}, for all eigenvalues \mc{\lm_{\scr i}} of \mc{A},\\
\tcr{or:} \mc{1} is a \tcr{regular, dominant}  eigenvalue of \mc{A}. 
   
\item[\tcr{Theorem 7:}] If \mc{|\lm_{\scr i}| < 1} \tcg{for every} eigenvalue \mc{\lm_{\scr i}} of \mc{A}, then \mc{A} is power convergent and 
\tcb{$$
\lim_{n\ra\infty\rule{0pt}{15pt}} A^n = 0.\vspace{-35pt}
$$} 

\item[\tcr{Theorem 8:}] If \mc{\lm_{\scr1} = 1} is a \tcg{regular, dominant}  eigenvalue of \mc{A}, then \mc{A} is power convergent and 
\tcb{$$
\lim_{n\ra\infty\rule{0pt}{15pt}} A^n = E_{\scr10} \ne 0,
$$}
where \mc{E_{\scr10}} denotes the lowest order \tcr{constituent matrix} of \mc{A} which is associated to the eigenvalue \mc{\lm_{\scr1} = 1}. Moreover, the other constituent matrices associated to \mc{\lm_1} are all zero:
\tcb{$$
E_{\scr11} = E_{\scr12} = \ldots = E_{\scr1m_1-1} = 0.
$$} 
\end{description} 
\end{titlepage}
\end{document}                          