<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% m211og91.tex - Limits of matrices  - new 17/18
%  
\newcommand{\titlesize}{30pt} % for PcTeX fonts -usual: 30pt 
\input{m211oh.tex}
%
\newcommand{\bmat}{\left(\begin{array}}
\newcommand{\emat}{\end{array}\right)}
%
\begin{titlepage}
\vspace*{-30pt}
\begin{center}\tcr{\rmH Limits of Matrix Sequences} 
\end{center} %\vspace{0pt}
%\LARGE              
\huge

\begin{description}  
\item[\tcr{Definition:}] Let \mc{A_1, A_2, \ldots} be a sequence of \mc{m\times n} matrices,
and write 
$$
\tcb{A_k \;=\; \bmat{ccc}
a_{11}^{(k)} &amp; \ldots &amp; a_{1n}^{(k)} \\
\vdots &amp;  &amp; \vdots \\
a_{m1}^{(k)} &amp; \ldots &amp; a_{mn}^{(k)} \\
\emat.}
$$
We say that the \emph{\tcr{limit}} \mc{A} of this sequence \emph{\tcr{exists}} and write
$$
\tcb{\lim_{k\ra\infty} A_k \;=\; A,}
$$
if \mc{A = (a_{ij})} is an \mc{m\times n} matrix such that 
$$
\tcb{\lim_{k\ra\infty} a_{ij}^{(k)} \;=\; a_{ij},} \quad\mbox{for all } 
\tcb{1\le i\le m, 1\le j \le n.}  
$$

\item[\tcr{Theorem 3 (Rules for limits):}] Suppose that \mc{A = \lim_{k\ra\infty} A_k} exists 
and that \mc{B, C} are matrices such that the matrix product \mc{CA_kB} is defined. Then: 

\tcr{\bf(a)} \mc{\displaystyle\lim_{k\ra\infty} A_kB  \; = \; AB.}

\tcr{\bf(b)} \mc{\displaystyle\lim_{k\ra\infty} CA_k  \; = \; CA.}

\tcr{\bf(c)} \mc{\displaystyle\lim_{k\ra\infty} CA_kB  \; = \; CAB.}
\end{description} 
\end{titlepage}
\end{document}                          </pre></body></html>