% m211oh96.tex (prev.: *oh92.*) - Geometric Series of Matrices
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\begin{center}\tcr{\rmH Geometric Series of Matrices} \end{center} 
\huge

\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
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\begin{description}
\item[\tcr{Definition:}] Let \mc{T} be any square matrix. Then the sequence \mc{\{S_n\}_{\scr n\ge0}} defined by
\tcb{$$
S_{\scr n} = I + T + \ldots + T^{\scr n-1},\quad S_{\scr 0} = I,
$$} 
is called the \tcr{geometric series generated by} \mc{T}. The series \tcr{converges} if the sequence \mc{\{S_n\}_{\scr n\ge0}} converges; we then write
\tcb{$$
\sum_{\scr n=0}^{\scr \infty} T^{\scr n} = \lim_{\scr\rule{0pt}{12pt} n\ra\infty} S_n.
$$}\vspace{-25pt}   
\item[\tcr{Theorem 11:}] The \tcg{geometric series generated by} \mc{T} converges if and only if 
\tcb{\beq
|\lm_{\scr i}| < 1,\; \mbox{for each eigenvalue}\,\, \lm_i\,\, \mbox{of}\,\, T. \label{1}
\eeq}
If this condition holds, then \mc{I - T} is \tcg{invertible} and we have
\tcb{\beq
S_{\scr n} := \sum_{\scr k=0}^{\scr n-1} T^{\scr k} = (I- T)^{\scr-1}(I - T^{\scr n}),
\eeq}
and hence the series converges to 
\tcb{\beq
  \sum_{\scr k=0}^{\scr \infty} T^{\scr k} = (I- T)^{\scr-1}.\vspace{-20pt}
\eeq}
\item[\tcr{Remark:}] Note that condition (\ref{1}) is equivalent to:
\tcb{\beq
\lim_{\scr\rule{0pt}{12pt}n\ra\infty} T^n\; = \; 0.
\eeq}
\end{description} 
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