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\begin{center}\tcr{\rmH %Evaluating Matrix Polynomials - 
Comparison of the Three Methods} \end{center} %\vspace{0pt}

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\begin{description}  

\item[\tcr{Method I:}] based on the \tcr{Jordan Canonical Form}; cf.\ \tcg{Theorem 5}. \vspace{-10pt}  

-- find \mc{P} such that \mc{J = P^{-1}AP} is a \vspace{-3pt} 
$\left\{\begin{array}{l} \mbox{\tcr{diagonal}} \\ \mbox{\tcr{Jordan}}\end{array}\right\}$
matrix; then \mc{f(A) = Pf(J)P^{-1}}. (Use the formula of \tcg{Theorem 7} to compute \mc{f(J)}.)

-- \tcr{good} for \tcg{diagonable} matrices, but usually \tcr{not practical} for \tcg{non-diagonable} matrices.

\item[\tcr{Method II:}] based on the \tcr{Cayley-Hamilton Theorem;} cf.\ \tcg{Theorem 8}.  

-- find \mc{r(t) = \mbox{rem}(f, \ch_A)}; then \mc{f(A) = r(A)}. (To compute \mc{r(t)}, use \tcg{long division} (if possible) or the \tcg{method of substitution} or the \tcg{remainder formula}.)
 
- \tcr{drawback:} one still has to compute \mc{r(A)} by the naive method.

\item[\tcr{Method III:}] based on the \tcr{Spectral Decomposition Theorem;} cf.\ \tcg{Theorem 12}.  

-- find the \tcr{constituent matrices} \mc{E_{ik}} by substituting suitable divisors of \mc{\ch_A(t)} into 
the \tcg{spectral decomposition formula}.

-- usually the \tcr{best} method.

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