% m211oh87.tex (prev.: *oh83.*) - The Invariance Property
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\begin{center}\tcr{\rmH The Invariance Property} \end{center} %\vspace{0pt}
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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}}
\newcommand{\bmat}{\left(\begin{array}}
\newcommand{\emat}{\end{array}\right)}

\begin{description}  

\item[\tcr{Theorem 3:}] \tcg{(Invariance Property)} If \mc{B = P^{-1}AP}, then\vspace{-10pt}
\tcb{\beqn
\ch_{A}(t) & = & \ch_B(t),\\ 
E_A(\lm) & = & PE_B(\lm)\; :=\; \{P\vec{v} : \vec{v} \in E_B(\lm)\}.
\eeqn}
In particular, we have
\tcb{\beqn
   m_A (\lm) & = & m_B(\lm), \\
   \nu_A(\lm) & = & \nu_B(\lm).\vspace*{-5cm} 
\eeqn} \vspace*{-1cm}

\item[\tcr{Example.}] $\begin{array}[t]{rl}
\mbox{Let} & \tcb{B = \bmat{ccc}
2 & 1 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2 
\emat,\; P = \bmat{ccc}
2 & 1 & 1 \\
1 & 1 & 1 \\
2 & 0 & 1 
\emat},\\
\mbox{and} &
\tcb{A = PBP^{\scr-1} = \bmat{ccr}
4 & 0 & -2 \\
1 & 2 & -1 \\
2 & 0 & 0 
\emat}.
\end{array}$

\exprow{1.3}
\mc{\begin{array}[t]{rrcl}
\tcg{\mbox{Then:}} 
& E_B(2) & \stackrel{\scr Th.2}{=} & \tcb{\{c_{\scr1}(1,0)^{\scr t}\}\oplus\{c_{\scr2}(1)\}} \\
     &    & = & \{c_{\scr1}(1,0,0)^{\scr t} + c_{\scr2}(0,0,1)^{\scr t}\}\\
\Rightarrow & E_A(2) & \stackrel{\scr Th.3}{=} &
\{c_{\scr1}\underbrace{P(1, 0, 0)^{\scr t}}_{\parbox{3.0cm}{\Large$1^{st}$ column\vspace{-5pt}\\ \hspace*{1cm}of $P$}}
+c_{\scr2}\underbrace{P(0, 0, 1)^{\scr t}}_{\parbox{3.50cm}{\Large$3^{rd}$ column\vspace{-5pt}\\ \hspace*{1cm}of $P$}}\}
 \\
& & = & \{c_{\scr1}(2,1,2)^t + c_{\scr2}(1,1,1)^t\}. 
\end{array}}
\end{description} 
\end{titlepage}
\end{document}                          