% m211oh76.tex {prev.: oh73.*) - Discrete Linear Systems (revised 03) 
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\begin{center}\tcr{\rmH Discrete Linear Systems}
%\vspace{5pt}\\ and Difference Equations} 
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\begin{description}  

\item[\tcr{Definition:}]  A \tcr{\em discrete linear system} \tcg{(d.l.s.)} is a system of equations of the form 
$$
\tcb{\vec{u}_{n+1} = A_n \vec{u}_n+\vec{b}_n}
$$ 
where \mc{n=0,1,2,\dots}. Here the vectors 
\mc{\vec{b}_0,\vec{b}_1,\dots \in \C^m} 
and the matrices \mc{A_n} (all of size \mc{m \times m}) are \tcg{given}, and we want to \tcg{solve} for the vectors \mc{\vec{u}_n}
in terms of \mc{\vec{u}_0}. 

 The system is said to have \tcr{\em constant coefficients} if \mc{A_n=A} is independent of \mc{n}. Moreover, it is called \tcr{\em homogeneous} if \mc{\vec{b}_n=\vec{0}} for all \mc{n}.

\iffalse
\item[\tcr{Definition:}]  A \tcr{\em difference equation of order} \mc{k} is an equation of the form: 
$$
\tcb{a_{n+k}=b_n+c_0a_n+a_1a_{n+1}+\dots+c_{k-1}a_{n+k-1},} $$ 
where \mc{n=0,1,\dots}. Here the numbers \mc{c_0,c_1,\dots,} \mc{c_{k-1} \in \C} and \mc{b_0,b_1,\dots,b_n,\dots} are \tcg{given} and we 
want to \tcg{solve} for the \mc{a_n}'s (usually in terms of certain \tcg{initial values} \mc{a_0,a_1,\dots,a_{k-1} \in \C}).

\item[\tcr{Note:}] Every \tcg{difference equation} defines a \tcg{d.l.s.}! 
\fi
\item[\tcr{Theorem 13:}] The \tcr{unique} solution of a \tcr{homogeneous} \tcg{discrete linear system} \mc{\vec{v}_{n+1} = A\vec{v}_n} is given by
$$
\tcb{\vec{v}_n = A^n\vec{v}_0.}
$$

\item[\tcr{Corollary:}] If \mc{\lm_1,\ldots,\lm_s} are the distinct \tcg{eigenvalues} of \mc{A} with \tcg{multiplicities}  \mc{m_1, \ldots, m_s}, then we have
$$
\tcb{\vec{v}_n = \sum_{i=1}^s \sum_{k=0}^{m_i-1} k!{n\choose k} \lm_i^{n-k}\vec{u}_{ik},}
$$
where the vectors \mc{\vec{u}_{ik} = E_{ik}\vec{v}_0} do not depend on \mc{n}.
\end{description}
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