exercise:Ed0d60ec05: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> The theorem that <math>\mat {P}^n \to \mat {W}</math> was proved only for the case that <math>\mat{P}</math> has no zero entries. Fill in the details of the following extension to the case that <math>\mat{P}</math> is regular. Since <math>\mat{...") |
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\newcommand{\mathds}{\mathbb}</math></div> The theorem that <math>\mat {P}^n \to \mat {W}</math> | \newcommand{\mathds}{\mathbb}</math></div> The theorem that <math>\mat {P}^n \to \mat {W}</math> was proved only for the case that <math>\mat{P}</math> has no zero entries. Fill in the details of the following extension to the case that <math>\mat{P}</math> is regular. Since <math>\mat{P}</math> is regular, for some <math>N, \mat {P}^N</math> has no zeros. Thus, the proof given shows that <math>M_{nN} - m_{nN}</math> approaches 0 as <math>n</math> tends to infinity. However, the difference <math>M_n - m_n</math> can never increase. (Why?) Hence, if we know that the differences obtained by looking at every <math>N</math>th time tend to 0, then the entire sequence must also tend to 0. | ||
was proved only for the case that <math>\mat{P}</math> has no zero entries. Fill in the | |||
details | |||
of the following extension to the case that <math>\mat{P}</math> is regular. Since | |||
<math>\mat{P}</math> is | |||
regular, for some <math>N, \mat {P}^N</math> has no zeros. Thus, the proof given shows | |||
that | |||
<math>M_{nN} - m_{nN}</math> approaches 0 as <math>n</math> tends to infinity. However, the | |||
difference <math>M_n - m_n</math> can never increase. (Why?) Hence, if we know that the | |||
differences obtained by looking at every <math>N</math>th time tend to 0, then the entire | |||
sequence must also tend to 0. | |||
Latest revision as of 22:15, 15 June 2024
[math]
\newcommand{\NA}{{\rm NA}}
\newcommand{\mat}[1]{{\bf#1}}
\newcommand{\exref}[1]{\ref{##1}}
\newcommand{\secstoprocess}{\all}
\newcommand{\NA}{{\rm NA}}
\newcommand{\mathds}{\mathbb}[/math]
The theorem that [math]\mat {P}^n \to \mat {W}[/math] was proved only for the case that [math]\mat{P}[/math] has no zero entries. Fill in the details of the following extension to the case that [math]\mat{P}[/math] is regular. Since [math]\mat{P}[/math] is regular, for some [math]N, \mat {P}^N[/math] has no zeros. Thus, the proof given shows that [math]M_{nN} - m_{nN}[/math] approaches 0 as [math]n[/math] tends to infinity. However, the difference [math]M_n - m_n[/math] can never increase. (Why?) Hence, if we know that the differences obtained by looking at every [math]N[/math]th time tend to 0, then the entire sequence must also tend to 0.