%m211oh64.tex - Matrix Polynomials - first properties (new 03)
%
\newcommand{\titlesize}{30pt} % for title -usual: 30pt 
\input{m211oh.tex}
%
\begin{titlepage}
\vspace*{-20pt}

\begin{center}\tcr{\rmH Matrix Polynomials} \end{center} %\vspace{0pt}
%\LARGE              
\huge

\begin{description}  

\item[\tcr{Notation:}] Let \mc{A \in M_m(\C) = } set of all \mc{m\times m} matrices with entries in \mc{\C} and let %\mc{f(t)\in\C[t]} be a polynomial,
$$
 \tcb{f(t)=c_0 + c_1t + c_2t^2 + \ldots + c_nt^n\in \C[t]} 
$$
be a complex polynomial. 
Then \mc{f(A)} denotes the \tcg{matrix expression}
$$
\tcb{f(A) = c_0I + c_1A + c_2A^2 + \ldots + c_nA^n \in M_{m}(\C).}
$$

\item[\tcr{Theorem 1:}]  If \vspace{-20pt} 
$$
\tcb{A\; =\; \mbox{\rm Diag }(a_1,a_2, \dots,a_m)\; %\stackrel{\text{def.}}
{=}\; 
%\begin{pmatrix}
\left(\begin{array}{cccc}
a_1&0&\dots&0\\
0&a_2&\ddots&\vdots\\
\vdots&\ddots&\ddots&0\\
0&\dots&0&a_m
%\end{pmatrix}
\end{array}\right)}
$$
is a \tcg{diagonal matrix}, then for any %polynomial 
\mc{f(t) \in \C[t]} we have:
$$
\tcb{f(A) \, = \, \mbox{\rm Diag} (f(a_1),f(a_2), \dots,f(a_m)).}
\iffalse
\begin{array}{cccc}
f(a_1)&0&\dots&0\\
0&f(a_2)&\ddots&\vdots\\
\vdots&\ddots&\ddots&0\\
0&\dots&0&f(a_m)
%\end{pmatrix}.
\end{array}
\fi
$$

\item[\tcr{Definition:}] 
%Two $m \times m$ matrices $A$ and $B$ are called \tcr{similar} if 
%If \mc{B=P^{-1}AP} for some invertible matrix \mc{P \in M_{m}(\C)},
%then \mc{B} is called \tcr{similar} to \mc{A}.
Two matrices \mc{A} and \mc{B} are called \tcr{similar} if 
\mc{B=P^{-1}AP} for some \tcg{invertible} matrix \mc{P}. % \in M_{m}(\C)}.


\item[\tcr{Theorem 2:}] 
Let \mc{A} and \mc{B=P^{-1}AP} be two \tcg{similar matrices}. Then:
\begin{eqnarray*}
  \tcb{A^n}  &\tcb{=}& \tcb{PB^nP^{-1},\quad \mbox{for all $n \geq 0$,}} \\
  \tcb{f(A)} &\tcb{=}& \tcb{Pf(B)P^{-1},\quad \mbox{for all $f \in \C[t]$}.}
\end{eqnarray*}
   
\end{description}
\end{titlepage}

\end{document}                          