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\begin{center}\tcr{\rmH Evaluating Matrix \vspace{5pt}Polynomials:\\ Method I (Diagonable Case)} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcr{Given:}] an \mc{m\times m} matrix \mc{A}:
 
\item[\tcr{1)}] Verify that \mc{A} is \tcg{diagonable} by checking whether any one of the following criteria applies:

  $\bullet$ Is \mc{A} a \tcr{real symmetric} matrix? 

  $\bullet$ Does \mc{A} have \mc{m} \tcr{distinct} \tcg{eigenvalues}? (Factor the characteristic polynomial \mc{\mbox{ch}_A(t)}.)

$\bullet$  Do the \tcg{eigenvectors} of \mc{A} form a \tcr{basis} of \mc{\C^m}? 

\item[Note:] If \tcr{none} of the above criteria apply, then \mc{A} is \tcg{not diagonable} and this method cannot be used.

\item[\tcr{2)}] Compute the \tcg{eigenvalues} and the corresponding \tcg{eigen\-spaces} of \mc{A} and hence find a basis of \mc{\C^m} consisting of eigenvectors \mc{\vec{v}_{\scr1},\ldots, \vec{v}_{m}} of \mc{A}, with corresponding eigenvalues \mc{\lm_{\scr1}, \ldots, \lm_m.}  Then the matrix \mc{P = (\vec{v}_{\scr1}| \dots |\vec{v}_m)} is invertible and has the property that \mc{A = PD P^{\scr -1}}, where 
\mc{D = \mbox{Diag}(\lm_{\scr1},\lm_{\scr2}, \ldots, \lm_m)}.

\item[\tcr{3)}] Use the following formula (cf.\ \tcg{Theorem 2, Corollary}) to calculate \mc{f(A)}:
$$
\tcb{f(A) = P\mbox{Diag}(f(\lm_{\scr1}), f(\lm_{\scr2}), \ldots, f(\lm_{m}))P^{-1}.}
$$
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