% m211oh69.tex (prev.: oh65) - The Jordan Canonical Form 
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\begin{center}\tcr{\rmH The Jordan Canonical Form} \end{center} 
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\begin{description}  

\item[\tcr{Definition:}] A \tcg{block diagonal} matrix 
$$
\tcb{J\; =\; \mbox{Diag}(J_{\scr1}, J_{\scr2}, \ldots, J_{\scr r})} 
$$
determined by \tcg{Jordan blocks} \mc{J_{\scr i} = J(\lm_{\scr i}, m_{\scr i})} is called a \tcr{Jordan matrix}. 

\item[\tcr{Theorem 5:}] 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{$$
P^{\scr-1}AP = \mbox{Diag}(J_{\scr1}, \ldots, J_{\scr i}, \ldots)
$$}
is a block diagonal matrix consisting of certain \tcg{Jordan blocks} \mc{J_{\scr i} = J(\lm_{\scr i}, k_{\scr i})} associated to the eigenvalues of \mc{A}. These blocks are \tcr{uniquely determined} by \mc{A} (up to order).
\iffalse
 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}}.
\fi
\item[\tcr{Historical Remark:}] The terms \tcg{Jordan block, Jordan matrix} etc.\ honour \tcr{Camille Jordan} \tcg{(1838 -- 1922)}, who in \tcg{1870} published the influential book \tcr{``Trait\'e des substitutions et des \'equations alg\'ebriques''} (667 pages). Van der Waerden (in his ``History of Algebra'') calls this book \tcg{``a masterpiece of mathematical architecture''}.
The above theorem is found (in essence) in book 2, chapter 3 of Jordan's Trait\'e.


\end{description} 
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