% m211oh66.tex  - Diagonalization Theorems (new Feb 03)
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\begin{center}\tcr{\rmH Diagonalization Theorems} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcr{Theorem 3}] \tcg{(Diagonalization Theorem)} \\
\tcr{(a)} An \mc{m\times m} matrix \mc{A} is \tcr{diagonable} if and only if 
\mc{A} has \mc{m} \tcr{linearly independent eigenvectors}. 
%(i.e.\ if and only if \mc{\C^m} has a \tcr{basis} consisting of %eigenvectors 
of \mc{A}.

\tcr{(b)} Suppose \mc{\vec{v}_1,\dots,\vec{v}_m \in \C^m} is a 
\tcg{linearly independent} set of \tcg{eigenvectors} of \mc{A} with 
corresponding eigenvalues \mc{\lambda_1, \dots, \lambda_m} 
(so \mc{A\vec{v}_i =\lambda_i\vec{v}_i}, for \mc{1 \le i \le m}). Then the matrix
\mc{P = (\vec{v}_1|\dots|\vec{v}_m)} 
%(whose column vectors are \mc{\vec{v}_1,\ldots,\vec{v}_m}) 
is \tcg{invertible} and we have 
$$
%\tcb{AP = P\mbox{Diag}(\lm_1,\ldots,\lm_m)\quad\mbox{or, equivalently,} 
\tcb{P^{-1}AP = \mbox{Diag}(\lm_1,\ldots,\lm_m).}
$$ 

\item[\tcr{Remark:}] Write 
$$
\tcb{\mbox{ch}_{A}(t) = (t -\lm_1)^{m_1}\cdots (t-\lm_r)^{m_r},} 
$$
where the \mc{\lm_i}'s are \tcr{distinct}, and let \mc{B_i}
be a \tcg{basis} of the corresponding eigenspaces \mc{E_A(\lm_i)},
\mc{1 \le i \le r}.
Then it can be shown that 
%\mc{B = B_1 \cup B_2 \cup \ldots \cup B_r} 
$$
\tcb{B = B_1 \cup B_2 \cup \ldots \cup B_r} 
$$
is a \tcg{linearly independent} set. Thus: 

$\tcb{A} \mbox{ is \tcr{diagonable}}\; \tcb{\Lra}\; 
\tcb{B} \mbox{ is a \tcr{basis} of } \tcb{\C^m} 
\; \tcb{\Lra}\; \tcb{\#B = m}$ \\
\hspace*{100pt}$\tcb{\Lra} \; 
\tcb{\dim E_A(\lm_1) + \ldots +  \dim E_A(\lm_r) =m.}$ 
%\end{eqnarray*}


\item[\tcr{Corollary:}] If \mc{A} has \mc{m} \tcr{distinct} eigenvalues 
(i.e.\ if \mc{\mbox{ch}_A(t)} has \mc{m} \tcr{distinct} roots), then \mc{A} is 
\tcr{diagonable.}

\item[\tcr{Warning:}] The \tcg{converse} of this corollary is \tcr{false}:\\ 
a matrix \mc{A} can be \tcg{diagonable} \tcr{yet} have \tcg{repeated} eigenvalues.

\tcr{Example:} the \mc{m\times m} identity matrix \mc{I} is diagonal 
(hence \tcg{diagonable}), but has only  \tcr{one} eigenvalue \mc{\lm_1 =1} 
(repeated \mc{m} times). 

\item[\tcr{Remark:}]
If \mc{A} is a \tcg{real} matrix (i.e.\ all entries of \mc{A} are real numbers) and 
is \tcr{symmetric}, i.e.\ \mc{A^t = A}, then \mc{A} is \tcg{automatically diagonable,} 
as the the following useful result shows:
%which is often useful for deciding \tcg{diagonalizability}: 

\item[\tcr{Theorem 4}] \tcg{(Principal Axis Theorem)}  \\
If \mc{A} is a \tcr{real symmetric} matrix, then \mc{A} is 
\tcr{orthogonally diagonable}; in other words, there exists an \tcg{orthogonal} 
matrix \mc{P} (i.e.\ a real matrix satisfying 
\mc{P^{-1} = P^t}) such that \mc{P^{-1}AP} is a diagonal matrix.

\item[\tcr{Remark:}] The \tcg{name} of this theorem comes from the fact 
that this theorem can used to show that \tcr{quadrics} in \mc{\R^n} 
centered at the origin
(e.g.\ \tcg{ellipses} in \mc{\R^2}, \tcg{ellipsoids} in \mc{\R^3}, etc.) 
can be \tcg{rotated} so that their \tcr{principal axes} are along the 
\tcg{coordinate axes} of \mc{\R^n}.   
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