<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% m211oh71.tex (prev.: oh67) - Evaluating Matrix Polynomials - Method II 
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\begin{center}\tcr{\rmH Evaluating Matrix\vspace{10pt} Polynomials:\\ Method II} \end{center} %\vspace{0pt}
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\newcommand{\HH}{{\mathbb H}}
\newcommand{\OO}{{\mathbb O}}

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\item[\tcr{Theorem 8}] \tcg{(Cayley-Hamilton).} For any square matrix \mc{A} we have:
$$
\tcb{\ch_A(A) = 0.}\vspace{5pt}
$$

\item[\tcr{Historical Notes:}] \rule{0pt}{5pt}

\tcr{William Rowan Hamilton} \tcg{(1805 - 1865)}:
invented in \tcr{1843} the \tcg{quaternions} (or \tcg{Hamiltonians})
\mc{\HH}. 

\tcr{Arthur Cayley} \tcg{(1821 -- 1895)}: invented in \tcr{1845} the \tcg{octonians} (or \tcg{Cayley numbers}) \mc{\OO}.

\item[\tcr{Note:}] We have the \tcr{hierarchy}: \mc{\R \subset \C \subset \HH \subset \OO}.

 \vspace{10pt}

\item[\tcr{Method II:}] (of evaluating matrix polynomials)

\tcr{Given:} an \mc{m\times m} matrix \mc{A} and a polynomial \mc{f(t)}.

\tcr{Compute:} \mc{f(A)}.
 
\tcr{1)} Calculate the characteristic polynomial \mc{\ch_A(t)}.

\tcr{2)} Compute \mc{r(t) = \mbox{rem}(f, \ch_A)}. [Use \tcg{long division} or the \tcg{substitution method}.]

\tcr{3)} Then \mc{f(A) = r(A)}, where \mc{r(A)} is evaluated the \tcg{naive way}.
 
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