
\subsection{\bf Orthonormal Eigenvectors and}
\subsection{\bf Unitary Transformation Matrices}

\begin{slide}

\begin{itemize}
\item We choose to normalize each eigenvector
\[
\langle v_k | v_k \rangle  =  \sum_{i=1}^n \left. v_k^* \right)_i
v_k \left. \right)_i =  1
\]

\item and we have proved
\[
\langle v_k | v_l \rangle = 0 \hspace{2em} l\neq k
\]

\item eigenvectors are ORTHOnormal whether or not eigenvalues are
degenerate

\item Define a matrix $U$ so that columns of $U^+$ are orthonormal
eigenvectors
\[
U^+ = \left\lbrack \begin{array}{cccccc}
\left. v_1 \right)_1 &&&&& \left. v_n \right)_1 \\
\left. v_1 \right)_2 \\
\left. v_1 \right)_3 \\
\\
\\
\left. v_1 \right)_n &&&&& \left. v_n \right)_n \end{array}
\right\rbrack
\]

\item $U$ is UNITARY \hspace{2em} $U^+ U = I$

\item This unitary condition is exactly the orthonormality condition.
\end{itemize}

\end{slide}

