% m211oh86.tex (prev.: *oh82.8) - Algebraic and Geometric Multiplicities
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\input{m211oh.tex}
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\begin{titlepage}
\vspace{-20pt}
\begin{center}\tcr{\rmH Algebraic and Geometric Multiplicities} \end{center} %\vspace{0pt}
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\huge

\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{Notation:}] Let \mc{A} be an \mc{m \times m} matrix and let \mc{\lm \in \C}. Then the \tcr{algebraic multiplicity} and  \tcr{geometric multiplicity} of \mc{\lm} in \mc{A} are respectively,

\mc{m_{\scr A}(\lm) = \mult_{\scr\lm}(ch_{\scr A}),} 

\mc{\nu_{\scr A}(\lm) = \dim_{\scr\C} E_{\scr A}(\lm) = m - \rank(A - \lm I)}.
 
\exprow{1.2}
\item[\tcr{Example:}]If \mc{A = J(\lm, m)} is a \tcg{Jordan block}, then \mc{m_{\scr A}(\lm) = m} and \mc{\nu_{\scr A}(\lm) = 1.}

\item[\tcr{Theorem 1:}] \tcg{(Sum Formula)} If \mc{A = \mbox{Diag}(B, C)}, \vspace{-35pt} then 
\iffalse
\begin{tabular}{lc}
\parbox{3cm}{Then:\vspace{1.5cm}} &
\begin{minipage}[t]{9cm}\mc{\begin{array}{rcl}
m_{\scr A}(\lm) & = & m_{\scr B}(\lm) + m_{\scr C}(\lm), \\
\nu_{\scr A}(\lm) & = & \nu_{\scr B}(\lm) + \nu_{\scr C}(\lm).
\end{array} }
\end{minipage} \\
\parbox{4cm}{In fact:\vspace{3cm}} & \begin{minipage}[b]{9cm}
\mc{\begin{array}{rcl}
\ch_{\scr A}(\lm) & = & \ch_{\scr B}(\lm)\cdot \ch_{\scr C}(\lm), \\
E_{\scr A}(\lm) & = & E_{\scr B}(\lm)\bigoplus E_{\scr C}(\lm).
\end{array} }
\end{minipage}
\end{tabular}
\fi 
\tcb{\begin{eqnarray*}
m_{\scr A}(\lm) & = & m_{\scr B}(\lm) + m_{\scr C}(\lm), \\
\nu_{\scr A}(\lm) & = & \nu_{\scr B}(\lm) + \nu_{\scr C}(\lm).
\end{eqnarray*}}\vspace*{-35pt} 

\item[\tcr{Theorem 2:}] In fact, we have\vspace{-5pt} 
\tcb{\begin{eqnarray*}
\ch_{\scr A}(\lm) & = & \ch_{\scr B}(\lm)\cdot \ch_{\scr C}(\lm), \\
E_{\scr A}(\lm) & = & E_{\scr B}(\lm)\oplus E_{\scr C}(\lm).
\end{eqnarray*} }\vspace{-35pt}

\item[\tcr{Corollary:}] If \mc{J = \mbox{Diag}(J_{\scr 11}, J_{\scr 12}, \ldots, J_{\scr ij}, \ldots)} is a \tcg{Jordan matrix} with Jordan blocks \mc{J_{ij} = J(\lm_{\scr i}, k_{\scr ij})}, then
\begin{tabular}{rcl}
\mc{\nu_{\scr J}(\lm_{\scr i})}& \mc{=} & \#(Jordan blocks \mc{J_{\scr ij}} with e.value \mc{\lm_{\scr i}}), \\
\mc{m_{\scr J}(\lm_{\scr i})} & \mc{=} & sum of the sizes \mc{k_{\scr ij}} of the Jordan blocks \\
& & with eigenvalue \mc{\lm_{\scr i}}. 
\end{tabular}  
\end{description} 
\end{titlepage}
\end{document}                          