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\begin{center}\tcr{\rmH Evaluating Matrix Polynomials - Method III} \end{center} %\vspace{0pt}
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This method is based on:

\begin{description}  

\item[\tcr{Theorem 12}] \tcg{(The Spectral Decomposition Theorem):}\\
Let \mc{A} be an \mc{m \times m} matrix with eigenvalues \mc{\lm_{\scr1}, \ldots, \lm_{\scr s}} and multiplicities \mc{m_{\scr1}, \ldots, m_{\scr s}}; 
that is, %in other words, suppose that
\begin{equation}
 \tcb{\ch_A(t) = (t - \lm_{\scr1})^{\scr m_1}\ldots (t - \lm_{\scr s})^{\scr m_s}}. \label{1}
\end{equation} 
Then there exist \mc{m} %= m_{\scr1} + \ldots + m_{\scr s}} 
matrices \mc{E_{\scr ik}} 
(of size \mc{m \times m}) 
such that for every polynomial \mc{f(t) \in \C[t]} we have
\vspace{-10pt} 
\beq
\tcb{f(A) = \sum_{\scr i=1}^{\scr s} \sum_{\scr k=0}^{\scr m_i-1} f^{\scr(k)}(\lm_{\scr i})E_{ik}.}
\vspace{-10pt}
\eeq
\item[\tcr{Remarks:}] \tcr{1)} The matrices \mc{E_{\scr ik}} depend on \mc{A} but not on \mc{f}; they are called the \tcr{constituent matrices} of \mc{A}. As we shall see,\vspace{-10pt}
$$
\tcb{E_{\scr ik} = e_{\scr ik}(A),}
$$
where the \mc{e_{\scr ik}(t)} are the constituent polynomials of \mc{\ch_A(t)} (cf.\ \tcg{Theorem 10}).   

\tcr{2)} The word \tcr{``spectral''} in the title of the theorem refers to the set \mc{spectrum(A) = \{\lm_{\scr1}, \ldots, \lm_{\scr s}\}}, the set of \tcg{eigenvalues} of \mc{A}. ($\ra$ \tcg{Physics}) 
\end{description}
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