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

\item[\tcr{Method I:}] \tcg{(Non-diagonable case)} Given a matrix \mc{A}:
 
\tcr{1)} Find the \tcr{eigenvalues} \mc{\lm_i}, the \tcr{Jordan blocks} \mc{J_i = J(\lm_{\scr i}, m_{\scr i})} and hence the associated \tcr{Jordan matrix} \mc{J = \mbox{Diag}(J_{\scr1}, J_{\scr2}, \ldots, J_{\scr r})}. 

\tcr{2)} Find a matrix \mc{P} such that \mc{A = PJ P^{\scr -1}}.

\tcr{3)} Use \tcg{Theorem 7} below to calculate \mc{f(J_{\scr i})}.

\tcr{4)} Apply \tcg{Theorem 6} to calculate \mc{f(A)}:
$$
\tcb{f(A) = P\mbox{Diag}(f(J_{\scr1}), f(J_{\scr2}), \ldots, f(J_{\scr r}))P^{\scr-1}.}
$$

\item[\tcr{Theorem 7:}] Let \mc{J = J(\lm, m)} be a Jordan block.
Then for any \mc{f \in \C[x]}:
\end{description}
\exprow{1.3}
$$
\tcb{f(J) = 
\left(\begin{array}{ccccc}
f(\lm) & f'(\lm) & \frac{\scr1}{\scr2}f''(\lm) & \ldots & \frac{1}{(m-1)!}f^{\scr(m-1)}(\lm) \\
0 & f(\lm) & f'(\lm) & \ldots & \frac{1}{(m-2)!}f^{\scr(m-2)}(\lm) \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots &  & \ddots& f(\lm) & f'(\lm) \\
0 &\ldots & \ldots&  0 & f(\lm) \\
\end{array}\right)}
$$ 
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