% m211oh88.tex (prev.: *oh84.*) - The Jordan Canonical Form 
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\begin{center}\tcr{\rmH The Jordan Canonical Form} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcr{Theorem 4:}] Every square matrix \mc{A} is \tcg{similar} to a \tcg{Jordan matrix}. In other words, there is an invertible matrix \mc{P} such that
\tcb{$$
P^{\scr-1}AP = \mbox{Diag}(J_{\scr11}, \ldots, J_{\scr ij}, \ldots)
$$}
is  block diagonal matrix consisting of \tcg{Jordan blocks} \mc{J_{\scr ij} = J(\lm_{\scr i}, k_{\scr ij})}. \tcr{Moreover:} the \mc{J_{\scr ij}} are \tcr{unique up to order}, and we have:

\tcr{1)} The \mc{\lm_{\scr1}, \ldots, \lm_{\scr s}} are the (distinct) eigenvalues of \mc{A}. 

\tcr{2)} The \tcr{number of blocks} \mc{J_{\scr i1}, J_{\scr i2}, \ldots, J_{\scr ij}, \ldots} with the same eigenvalue \mc{\lm_{\scr i}} equals the \tcr{geometric multiplicity} 
\mc{\nu_{\scr i} = \nu_A(\lm_{\scr i})}. 

\tcr{3)} The \tcr{sum of the sizes} \mc{k_{ij}} of the blocks \mc{J_{\scr i1}, J_{\scr i2}, \ldots} equals the \tcr{algebraic multiplicity} \mc{m_{\scr i} = m_A(\lm_{\scr i})}:
\tcb{$$
 k_{\scr i1} + k_{\scr i2} + \ldots + k_{\scr i\nu_i} = m_i.
$$}
In particular: \mc{\nu_{\scr i} \le m_{\scr i}}.

\item[\tcr{Corollary:}] Two \mc{m\times m} matrices \mc{A} and \mc{B} are \tcr{similar} (i.e., \mc{B = P^{\scr-1}AP},
for some \mc{P}) if and only if they have the same \tcg{Jordan canonical form} (up to order).     
\end{description}
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\end{document}                          