% m211oh90.tex  - Generalized Eigenvectors
%  - revision 17/18 of the 06/07 Version (prev.: *oh86.*)
%
\newcommand{\titlesize}{30pt} % for PcTeX fonts -usual: 30pt 
\input{m211oh.tex}
%
%\begin{titlepage}
\vspace*{-20pt}
\begin{center}\tcr{\rmH Generalized Eigenvectors} \end{center} %\vspace{0pt}
%\LARGE           
\huge

\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
\newcommand{\vv}[1]{\vec{v}_{\scr #1}}
\newcommand{\rank}{\mbox{rank}}
\newcommand{\mult}{\mbox{mult}}

\begin{description}  
\item[\tcr{Definition:}] Let \mc{A} be an \mc{m \times m} matrix and \mc{\lm\in\C.} 
A vector \mc{\vv{} \in \C^{\scr m}} is called a \emph{\tcr{generalized \mc{\lm}-eigenvector}} 
of \mc{A} \emph{\tcr{of order \mc{\le p}}} if %\vspace{-20pt}
$$
\tcb{   (A - \lm I)^{\scr p}\vv{} = \vec{0}.}
$$
If, in addition, \mc{ (A - \lm I)^{\scr p-1}\vv{} \ne \vec{0}}, then we say that 
\mc{\vv{}} has \tcr{(precise) order} (or degree) \mc{p}. The space
%\end{description}%\vspace{-30pt}
\begin{eqnarray*}
\tcb{E_{\scr A}^{\scr p}(\lm)} &\tcb{=}& \tcb{\mbox{Nullsp}((A - \lm I)^{\scr p})} \\
&\tcb{=}& \tcb{\{\vv{} \in \C^{\scr m}: (A - \lm I)^{\scr p}\vv{} = \vec{0}\}.}
\end{eqnarray*}
is called the \emph{\tcr{\mc{p}-th generalized \mc{\lm}-eigenspace}} of \mc{A,} and its dimension
$$
\tcb{\nu_A^p(\lm) := \dim E^p_A(\lm) = m - \mbox{rank}((A-\lm I)^p)} 
$$
is the \emph{\tcr{\mc{p}-th generalized geometric multiplicity}} of \mc{A} with respect to \mc{\lm.}

\item[\tcr{Remarks:}] 1) For \mc{p=1} we recover the usual eigenspace:
$$
\tcb{E^1_A(\lm) \; = \; E_A(\lm),}\; \mbox{ and so }\; \tcb{\nu_A^1(\lm) = \nu_A(\lm).}
$$

2) It is clear that \mc{E^p_A(\lm) \subset E^{p+1}_A(\lm), \forall p.} Thus
\tcb{$$
\{0\} \subset E_{\scr A}^{\scr1}(\lm) = E_{\scr A}(\lm) 
\subset E_{\scr A}^{\scr 2}(\lm) \subset \ldots E_{\scr A}^{\scr p}(\lm)\subset \ldots
$$}
%\hspace*{30pt}

%***
\iffalse
Thus, if we let \mc{\nu_A^p(\lm):= \dim_{\C} E_{\scr A}^{\scr p}(\lm)}, then
\tcb{$$\exprow{1.2}
 \begin{array}{l} 
0 < \nu_{\scr A}^{\scr1}(\lm) = \nu_{\scr A}(\lm) < \ldots < \nu_{\scr A}^{\scr k-1}(A) \\ 
\hspace*{16pt}< \nu_{\scr A}^{\scr k}(A)= \nu_{\scr A}^{\scr k+1}(A) = \ldots 
= m_{\scr A}(\lm).
\end{array}
$$}\vspace{-20pt}\exprow{1.0}

\item[\tcr{Properties:}] \tcr{\bf1)} \mc{E_{\scr C}^{\scr p}(\lm) 
= E_{\scr A}^{\scr p}(\lm)\oplus E_{\scr B}^{\scr p}(\lm)}, if \mc{C 
= \mbox{Diag}(A, B)}; in particular, \mc{\nu_{\scr C}^{\scr p}(\lm) 
= \nu_{\scr A}^{\scr p}(\lm) + \nu_{\scr B}^{\scr p}(\lm)}.

\tcr{\bf2)} If \mc{B = P^{\scr-1}AP}, then \mc{E^{\scr j}_{\scr A}(\lm) 
= PE_{\scr B}^{\scr j}(\lm)}; in particular,
\mc{\nu^{\scr j}_{\scr B}(\lm) = \nu_{\scr A}^{\scr j}(\lm)}, for all \mc{j \ge 0}.  

\tcr{\bf3)} \mc{\nu^{\scr p}_{A}(\lm) - \nu^{\scr p-1}_{A}(\lm)}
\mc{= \parbox[t]{8cm}{\expcol{-5pt}\begin{array}[t]{ll}
\#(&\mbox{\tcg{Jordan blocks}}\, J(\lm, k_{\scr ij}) \\
&\mbox{\tcg{of JCF}}\, J\,  \mbox{\tcg{of size}}\, k_{ij} \ge p).
\end{array}}}
\fi
%***

\item[\tcr{Example:}] If \mc{A = J(\lm, m)} is a Jordan block, then 
$$
\tcb{E^p_{J(\lm,m)} = \mbox{Nullsp}((A-\lm I)^p) = \lb\vec e_1,\ldots, \vec e_p\rb,} 
$$
if \mc{p \le m,} and \mc{E^p_{J(\lm,m)} = \C^m,} if \mc{p\ge m.} Thus
$$
\tcb{\nu_{J(\lm,m)}^p(\lm) \;=\; \left\{
\begin{array}{ll}
p & \mbox{if } p \le m, \\
m & \mbox{if } p \ge m. \\
\end{array}
\right.}
$$

\item[\tcr{Theorem 5 (Properties of gen.\ eigenvectors):}] 
\tcr{\bf(a)} If \mc{A = \mbox{Diag}(B, C)} and if \mc{p\ge1}, then  
$$
\tcb{E_{\scr A}^{\scr p}(\lm) \;=\; E_{\scr A}^{\scr p}(\lm)\oplus E_{\scr B}^{\scr p}(\lm)}; 
$$
in particular, \mc{\nu_{\scr A}^{\scr p}(\lm) 
= \nu_{\scr B}^{\scr p}(\lm) + \nu_{\scr C}^{\scr p}(\lm)}.

\tcr{\bf(b)} If \mc{A = PBP^{-1},} then 
$$
\tcb{E^{\scr q}_{\scr A}(\lm) \;=\; PE^{\scr q}_{\scr B}(\lm),\quad\mbox{for all } q\ge 1;}
$$
in particular, \mc{\nu_{\scr A}^{\scr q}(\lm) = \nu_{\scr B}^{\scr q}(\lm),} 
for all \mc{q\ge1.}

\tcr{\bf(c)} If \mc{J} is a Jordan canonical form of \mc{A}, then  
$$
\tcb{\nu^{\scr p}_{\scr A}(\lm) - \nu^{\scr p-1}_{\scr A}(\lm)}\,
\tcb{= \parbox[t]{8cm}{\expcol{-5pt}\begin{array}[t]{ll}
\#(&\mbox{\tcg{Jordan blocks}}\, J(\lm, k_{\scr ij}) \\
&\mbox{\tcg{of JCF}}\, J\,  \mbox{\tcg{of size}}\, k_{ij} \ge p).
\end{array}}}
$$
\tcr{\bf(d)} We have that
$$
\tcb{\nu^{\scr p}_{\scr A}(\lm) = \nu^{\scr p+1}_{\scr A}(\lm) \Ra 
\tcb{\nu^{\scr p}_{\scr A}(\lm) = \nu^{\scr p+q}_{\scr A}(\lm), \mbox{ for all } q\ge1.}
}
$$
\item[\tcr{Corollary:}] The numbers \mc{\nu^{\scr p}_{\scr A}(\lm_i)} determine the JCF of \mc{A}
(by taking second differences).
\newpage

\item[\tcr{Example 1:}] Let \mc{A} be a matrix 
with characteristic polynomial \mc{\ch_A(t)=(t-7)^5} and 
\tcg{generalized geometric multiplicities}
\tcb{\begin{eqnarray*} 
\nu_A^*(7) &=& (\nu_A^1(7),\nu_A^2(7),\nu_A^3(7),\dots)\\
&=&(2,4,5,5,5,\dots)
\end{eqnarray*}}
Find the associated \tcr{Jordan canonical form} \mc{J} of \mc{A}. 
\item[\tcr{Solution:}] By using \tcg{Theorem 5} (or its \tcg{Corollary}) 
we obtain:

\newcommand{\phs}{\phantom{s}}

\tcb{\LARGE
%\begin{minipage}{10.1cm}
$ \exprow{1.25}
\begin{array}{cccccccr}
\nu_A^4-\nu_A^3&=&5-5&=&0&\Rightarrow& 0 \mbox{ blocks of size} &\geq 4\\
\nu_A^3-\nu_A^2&=&5-4&=&1&\Rightarrow& 1 \mbox{ block\phs\ of size}&\geq 3\\
\nu_A^2-\nu_A^1&=&4-2&=&2&\Rightarrow& 2 \mbox{ blocks of size} &\geq 2\\
\nu_A^1-\nu_A^0&=&2-0&=&2&\Rightarrow& 2 \mbox{ blocks of size}&\geq 1
\end{array}$
%\end{minipage}
\begin{minipage}{4cm}
\vspace{-0.1cm}
$\exprow{1.3}
\begin{array}{l}
 \}\ 1 \mbox{ bl.\ size } 3\\
 \}\ 1 \mbox{ bl.\ size } 2\\
 \}\ 0 \mbox{ bl.\ size } 1
\end{array}$
\end{minipage}}

and hence \mc{J} has: 
\tcb{$\begin{array}[t]{rcl}
0&\mbox{blocks of size}& 1 \times 1\\
1&\mbox{block\phs\ of size}& 2 \times 2\\
1&\mbox{block\phs\ of size}& 3 \times 3\\
0&\mbox{blocks of size}& 4 \times 4 \mbox{ etc.}
\end{array}$}

\tcr{Thus:} \mc{J=\mbox{Diag}(J(7,3),J(7,2))=\left(\begin{array}{c}
\begin{array}{ccc|cc}
7&1&0&0&0\\
0&7&1&0&0\\
0&0&7&0&0\\ \cline{1-5}
0&0&0&7&1\\
0&0&0&0&7
\end{array}
\end{array}\right)} (up to order of the blocks).

%*******

\newcommand{\Diag}{\mbox{diag}}

\item[\tcr{Example 2:}] Find the Jordan canonical form of 
\exprow{.8}
$$
\tcb{A = \exprow{.8} \left( 
{\begin{array}{rrrrr}
7 & 0 & 0 & 1 & 0 \\
0 & 7 & 0 & 0 & 0 \\
1 & 0 & 7 & 0 & 0 \\
0 & 0 & 0 & 7 & 0 \\
0 & 1 & 0 & 0 & 7
\end{array}}
 \right).}
$$

\item[\tcr{Solution:}] {\bf Step 1:} Find the \tcg{characteristic} \rlap{\tcg{polynomial.}}
% of \mc{A.} 

Expanding \mc{\det(A-tI)} along the first row yields
$$
\tcb{\mbox{ch}_A(t) = (-1)^5\det(A-tI) = (t-7)^5.}
$$  
Thus, \mc{m_A(7) =5}. Moreover, since 
$$
\tcb{B = A -7I = (\vec e_3|\vec e_5|\vec0|\vec e_1|\vec0),}
$$
we have \mc{\nu_A(7) = 5 - \mbox{rk}(B) = 2 = \#\mbox{(Jordan blocks)}.}
% so there are \mc{2} Jordan blocks. 

{\bf Step 2:} Calculate the \tcg{generalized geometric multiplicities.}

Since 
%\mc{(\vec v_1|\vec v_2|\vec v_3|\vec v_4|\vec v_5)B = (\vec v_3|\vec v_5|0|\vec v_1|0)}, 
\mc{B(\vec v_1|\vec v_2|\vec v_3|\vec v_4|\vec v_5) = (B\vec v_1|B\vec v_2|B\vec v_3|B\vec v_4|B\vec v_5)} and since \mc{B\vec e_i = i}-th column of \mc{B}, we see that 
$$ 
\tcb{B^2 = (B\vec e_3|B \vec e_5|B\vec0|B\vec e_1|B\vec0) = (\vec0|\vec0|\vec0|\vec e_3|\vec0),}
%\quad B^3 = (0|0|0|0|0)}  
$$ 
and similarly
$$
\tcb{B^3 = B\cdot B^2 =  (B\vec0|B\vec0|B\vec0|B\vec e_3|B\vec0) 
= (\vec0|\vec0|\vec0|\vec0|\vec0),}
$$
and so \mc{B, B^2}, and \mc{B^3} have ranks \mc{3, 1,} and \mc{0} respectively. 
Moreover, clearly \mc{B^p =0,} for all \mc{p\ge3.}

Thus, 
since \mc{\nu_A^p(7) = 5 - \mbox{rank}(B^p)}, we see that the 
\tcg{generalized geometric multiplicities} are 
$$
\tcb{\nu_A^*(7) = (2,4,5,5,\ldots).}
$$
{\bf Step 3:} Find the Jordan blocks by the \tcg{method of second differences.}
\sps

\noi Since \mc{\ch_A(t) = (t-7)^5} and 
since the generalized geometric multiplicities are 
$$
\tcb{\nu_A^*(7) = (2,4,5,5,\ldots),}
$$  
we can conclude by \tcr{Example 1} that the Jordan canonical form \mc{J} of \mc{A}
is
$$
\tcb{J=\Diag(J(7,3),J(7,2))=\left(\begin{array}{c}
\begin{array}{ccc|cc}
7&1&0&0&0\\
0&7&1&0&0\\
0&0&7&0&0\\ \cline{1-5}
0&0&0&7&1\\
0&0&0&0&7
\end{array}
\end{array}\right).}
%\quad\mbox{(up to order)}.
$$
\end{description} 
%\end{titlepage}



\end{document}                          