<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% m211oh92.tex (prev.: *oh88.*) - Power Convergent Matrices (Special Cases)
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\begin{center}\tcr{\rmH Power Convergent Matrices\vspace{8pt} \\
(Special Cases)} \end{center} 
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\newcommand{\ueq}{\vec{u}_{\scr eq}}
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\begin{description}
\item[\tcr{Definition:}] A square matrix \mc{A} is called \tcr{power convergent} if \mc{\displaystyle\lim_{k\ra\infty} A^k} exists.
 
\item[\tcr{Theorem 4:}]
A \tcg{diagonal} matrix \mc{A = \mbox{Diag}(\lm_{\scr1},\ldots, \lm_{\scr m})} is power convergent if and only if for \tcg{each} \mc{i} with \mc{1\le i\le m} we have:
\begin{equation}
\tcr{\mbox{either}}\quad\tcb{|\lm_{\scr i}| &lt; 1}\quad
\tcr{\mbox{or}}\quad\tcb{\lm_{\scr i} = 1.}\label{1}
\end{equation}

\item[\tcr{Corollary.}] A \tcg{diagonable} matrix \mc{A} is power convergent if and only if  each \tcg{eigenvalue} \mc{\lm_{\scr i}} satisfies \sref{1}.

\item[\tcr{Theorem 5:}] A \tcg{Jordan block} \mc{J = J(\lm, m)} is power convergent if and only if
\begin{equation}
\tcr{\mbox{either}}\quad\tcb{|\lm| &lt; 1}\quad
\tcr{\mbox{or}}\quad\tcb{\lm = 1\;\mbox{and}\;m =1.}\label{2}
\end{equation}

\item[\tcr{Remark.}] The \tcg{proof} of \tcg{Theorem 5} uses the fact that
$$
\tcb{|\lm| &lt; 1} \quad
\tcr{\Rightarrow}
\quad \exprow{.75}
\tcb{\lim_{n\ra\infty\rule{0pt}{15pt}} \bmat{c} n \\ a\\\emat\lm^{\scr n-a} = 0,}\;\;\mbox{for all }\tcb{a \in \N}.\exprow{1.0}  
$$

\item[\tcr{Corollary.}] If \mc{|\lm| &lt; 1}, then 
\mc{\displaystyle \lim_{k\ra\infty} J(\lm, m)^k = 0}.

\end{description} 
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