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\begin{center}\tcr{\rmH Review of Diagonalization} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcr{Definition:}] 
\tcg{1)} \mc{\lm\in\C} is an \tcr{eigenvalue} of an \mc{m\times m} matrix \mc{A} if \mc{A\vec{v} = \lm\vec{v}} for some \mc{\vec{v} \ne \vec{0}}.

\tcg{2)} \mc{\vec{v} \in \C^m} is an \tcr{eigenvector} of \mc{A} \tcr{with respect to} \mc{\lm \in \C} if \mc{A\vec{v} = \lm\vec{v}}.

\tcg{3)} The set \mc{E_A(\lm) = \{\vec{v} \in \C^m: A\vec{v} = \lm\vec{v}\}}
of eigenvectors w.r.t.\ \mc{\lm} is called the \tcr{$\lm$-eigenspace} of \mc{A}.

\tcg{4)} The \tcr{characteristic polynomial} of \mc{A} is
\tcb{\beqn
\mbox{ch}_A(t) & = & \det(tI - A)\, = \, (-1)^m\det(A - tI)\\
  & = & t^{\scr m} + a_{\scr1}t^{\scr m-1} + \ldots + a_{\scr m}.
\eeqn}\vspace{-1.25cm}

\item[\tcr{Facts:}] \tcg{1)} \mc{\lm \in \C} is an eigenvalue of \mc{A} 
\tcr{$\Leftrightarrow$} \tcb{ch$_A(\lm) = 0$}.
\tcr{Thus:} \mc{A} has precisely \mc{m} eigenvalues (w.\ mult.'s). 

\tcg{2)} If \mc{A} has \mc{m} \tcr{distinct} e.val's, then \mc{A} is \tcg{diagonable}.

\tcg{3)} \mc{A} is \tcg{diagonable} if and only if \mc{\C^m} has a \tcr{basis} consisting of eigenvectors of \mc{A}.

\tcg{4)} If \mc{\vec{v}_{\scr1}, \vec{v}_{\scr2},\ldots, \vec{v}_m \in \C^{\scr m}} is a basis of eigenvectors (so \mc{A\vec{v}_{\scr1} = \lm_{\scr1}\vec{v}_{\scr1}, A\vec{v}_{\scr2} = \lm_{\scr2}\vec{v}_{\scr2},\ldots}),
then 
$$
\tcb{A\underbrace{(\vec{v}_{\scr1}|\vec{v}_{\scr2}|\ldots
|\vec{v}_{\scr m})}_{P}\; = \; (\vec{v}_{\scr1}|\vec{v}_{\scr2}|\ldots
|\vec{v}_{\scr m})\mbox{Diag}(\lm_{\scr1},\ldots,\lm_{\scr m}),}\vspace{-5pt}
$$
\tcr{or, equivalently:}\hspace{10pt}   
\mc{A = P \mbox{Diag}(\lm_{\scr1},\ldots,\lm_{\scr m})P^{\scr-1}.}  
\end{description}
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\end{document}                          