% m211oh94.tex (prev.: *oh90.*) - Simple Eigenvalues
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\begin{center}\tcr{\rmH Simple Eigenvalues} \end{center} 
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\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[\tcr{Definition:}] 
An eigenvalue \mc{\lm} of \mc{A} is called \tcr{simple} if 
its \tcg{algebraic multiplicity}
\mc{m_{\scr A}(\lm) = 1}.

\item[\tcr{Remark.}] Clearly, each \tcg{simple} eigenvalue is \tcg{regular}.
  
\item[\tcr{Theorem 10:}] If \mc{A} is \tcr{power convergent} and \mc{1} is a \tcg{simple}  eigenvalue of \mc{A}, then
$$
\tcb{\lim_{\scr\rule{0pt}{15pt}n\ra \infty} A^n\; =\; E_{\scr10}\; =\; \frac{1}{\underbrace{\vec{u}^t\vec{v}}_{\tcg{\mbox{\Large scalar}}}}
\underbrace{\vec{u}\vec{v}^t}_{\mbox{\tcg{\Large matrix}}}},
$$
where:
\begin{eqnarray*}
\tcb{\vec{u} \in E_A(1)}& \mbox{is any non-zero \mc{1}-eigenvector of } \tcb{A}, \\
\tcb{\vec{v} \in E_{A^t}(1)}& \mbox{is any non-zero \mc{1}-eigenvector of } \tcb{A^t}. 
\end{eqnarray*}

\item[\tcr{Remark.}] Since \mc{\det(C^t) = \det(C)} for any matrix \mc{C}, it follows that
$$
\tcb{\mbox{ch}_{A^t}(t) = \mbox{ch}_A(t)}.    
$$
\tcr{Thus:} \mc{A} and \mc{A^t} have the same eigenvalues with the same algebraic multiplicities. (In fact, \mc{A} and \mc{A^t} are \tcg{similar}, so they even have the same \tcg{Jordan canonical form}.)
   
\end{description} 
\end{titlepage}

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