
\subsection{\bf Eigenvalues of Hermitean Matrices---I}

\begin{slide}

\begin{itemize}
\item For Hermitean $A$, suppose
\begin{eqnarray*}
A v_k & = & \lambda_k v_k \\
\left. A_{ij} v_k \right)_j & = & \lambda_k \left. v_k \right)_i
\end{eqnarray*}

\item Multiply both sides by $\left. v_k \right)_i^*$ and sum over
$i=1\ldots n$

\item \begin{eqnarray*}
\langle v_k \vert A \vert v_k \rangle & = & \lambda_k \langle v_k
\vert v_k \rangle \ \mbox{or} \\
\sum_{i,j=1}^n \left. v_k \right)_i^* A_{ij} \left. v_k \right)_j
& = & \lambda_k \sum_i \left. v_k \right)_i^* \left. v_k \right)_i
\end{eqnarray*}

\item Write this $\mbox{LHS}\ = \lambda_k\, \mbox{RHS}$.  The right
hand side (RHS) is clearly real and nonzero
\begin{eqnarray*}
\mbox{RHS} & = & \sum_i \left\vert \left. v_k \right)_i \right\vert^2 \\
& = & \left\langle v_k \vert v_k \right\rangle 
\end{eqnarray*}
\end{itemize}

\end{slide}

