% m211og93.tex - Properties of Constituent Matrices  
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\begin{center}\tcr{\rmH Properties of Constituent Matrices} 
\end{center} \vspace{-5pt}
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\begin{description}  
\item[\tcr{Notation:}] If \mc{A} is an \mc{m\times m} matrix, let
$$
\tcb{\mbox{ch}_A(t) = (t-\lm_1)^{m_1}(t - \lm_2)^{m_2}\cdots (t-\lm_s)^{m_s}} 
$$
be the factorization of \mc{\mbox{ch}_A(t)} (\mc{\lm_i}'s distinct), and  
let \mc{\{E_{ik}\}} be the set of constituent matrices of \mc{A}.

For any \mc{\lm \in \C} and \mc{k \ge 0} put
$$
\tcb{E_{\lm,k}^A = \left\{\begin{array}{ll}
E_{ik} & \mbox{if \mc{\lm = \lm_i} and \mc{k\le m_i-1}} \\
0 & \mbox{otherwise.}  
\end{array}\right.}
$$

\item[\tcr{Theorem:}] Let \mc{\lm \in \C} and \mc{k \ge0.} Then: %we have

\tcr{\bf (a)} If \mc{A = \mbox{diag}(A_1,\ldots, A_r)}, then 
%for any \mc{\lm \in \C} and \mc{k \ge0} we have
\beq
\tcb{E_{\lm, k}^A = \mbox{diag}(E_{\lm,k}^{A_1}, \ldots E^{A_r}_{\lm, k}).}
\eeq

\tcr{\bf (b)} If \mc{B = P^{-1}AP}, then 
%for any \mc{\lm \in \C} and \mc{k \ge0} we have
\beq
\tcb{E_{\lm, k}^B = P^{-1}E_{\lm,k}^{A}P.}
%\quad\mbox{for any \mc{\lm \in \C} and \mc{k \ge0}.}
\eeq

\item[\tcr{Corollary 1:}] In the situation of part (a) we have
\beq
\hspace*{20pt}\tcb{E^A_{\lm, k} = 0} \;\; \mbox{for } 
\tcb{k \ge \max(m_{A_1}(\lm), \ldots, m_{A_r}(\lm)).} 
\eeq

\newpage

\item[\tcr{Application:}] The \tcr{maximum size} of a Jordan block. % in \mc{J.}

Let \mc{\lm \in \C}, and put
$$
 \tcb{t_A(\lm) := \max\{k: J(\lm, k)\mbox{ appears as a block in } J_A\}}, 
$$
where \mc{J_A} is the Jordan canonical form of \mc{A}.
    
\item[\tcr{Corollary 2:}] We have that
\beq
\label{4}
\tcb{E_{\lm, k}^A = 0 \quad \Leftrightarrow \quad k \ge t_A(\lm).}
\eeq
 
\item[\tcr{Remarks:}] \tcr{\bf1)} Recall that \mc{\lm} is a \tcr{regular} eigenvalue of 
\mc{A} \mc{\Leftrightarrow} \mc{t_A(\lm) = 1.}  Thus by Corollary 2 we have
$$
\tcb{\lm} \mbox{ is regular eigenvalue } \tcb{ \Rightarrow E^A_{\lm, k} = 0 
\mbox{ for } k \ge 1.}
$$ 

\tcr{\bf2)} The number \mc{t_A(\lm)} %can also be defined in terms of the 
is connected with the \tcr{generalized geometric multiplicities} \mc{\nu_A^p(\lm)} 
as follows: %in the following way:
\beq
\tcb{\nu_A^{k}(\lm) = \nu_A^{k+1}(\lm) \quad \Lra \quad k \ge t_A(\lm).}
\eeq
Thus, we have the following pattern (if \mc{t =t_A(\lm)}): 
$$
\tcb{0 < \nu_A^1(\lm) < \ldots < \nu_A^t(\lm) = \nu^{t+1}_A(\lm) = \ldots}
$$
\tcr{\bf3)} If we put 
$$
\tcb{\mu_A(t) = (t-\lm_1)^{t_A(\lm_1)} \cdots (t-\lm_s)^{t_A(\lm_s)},}
$$
then \mc{\mu_A(t) |\mbox{ch}_A(t)} and we have \mc{\mu_A(A) = 0} by the 
Spectral Decomposition Formula and \sref{4}. This refines the 
\tcr{Cayley-Hamilton Theorem.} 
\end{description}
\end{document}                          