% m211oh68.tex (prev.: oh64) - Jordan Blocks
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\begin{center}\tcr{\rmH Jordan Blocks} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcr{Definition:}] Given \mc{\lm \in \C} and \mc{m \in \N}, the \mc{m \times m} matrix \vspace{-30pt}
%\exprow{1.1}
$$
\tcb{J(\lm, m)  = 
\left(\begin{array}{cccccc}
\lm & 1 & 0 & \cdots & \cdots &0 \\
0 & \lm & 1 &  0 & \cdots & 0 \\
\vdots  & \ddts  & \ddts & \ddts &\ddts& \vdots \\
\vdots &  & \ddts & \ddts & \ddts & 0 \\ 
\vdots & & & \ddts & \lm & 1 \\
0 & \cdots & \cdots & \cdots &  0 & \lm
\end{array}\right) }
$$ 
%\mc{J(\lm, m)} 
is called the \tcr{Jordan block} of size \mc{m\times m} associated to \mc{\lm \in \C}.

\item[\tcr{Notes:}] \tcr{1)} We have
\end{description} \vspace{-40pt}
$$
\tcb{\mbox{ch}_{\scr J(\lm, m)}(t) = (t - \lm)^m}
\mbox{ and }
\tcb{E_{\scr J(\lm, m)} = \{c\left(\begin{array}{c} 1 \\ 0 \\
 \vdots \\  0 \end{array}\right): c \in \C\}}.
$$
Thus, \mc{J(\lm, m)} is \tcr{not} diagonable (if \mc{m \ge 2}).  
%\end{description}

\tcr{2)} The matrices \mc{J(\lm, m)} are the \tcg{``building blocks''} of all \tcr{non-diagonable} matrices (up to similarity).

\tcr{3)} The only obstruction to diagonalizing a matrix is the presence
(up to similarity) of \tcg{Jordan blocks}.

\tcr{4)} Every matrix can be put into a \tcg{``generalized diagonal form''}, called the \tcr{Jordan canonical form}. This is the \tcg{closest} that a matrix can come to being \tcg{diagonable}.
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