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

\begin{slide}

\begin{itemize}
\item On the other hand, complex conjugate equation,
\begin{eqnarray*}
\mbox{LHS} & = & \sum_{i,j=1}^n \left. v_k \right)_i^* A_{ij} \left.
v_k \right)_j\\
\mbox{LHS}^* & = & \sum_{i,j=1}^n \left. v_k \right)_i 
\underbrace{A_{ij}^*}_{\mbox{as\ hermitean}\ \ A^*_{i,j} =  A_{ji}}
\left. v_k \right)_j^* \\
& = & \sum_{i,j=1}^n \left. v_k \right)_i A_{ji} \left. v_k \right)_j^*
\end{eqnarray*}
 
\item Now Interchange $i$ and $j$ labels
\begin{eqnarray*} 
\mbox{LHS*} & = & \sum_{i,j=1}^n \left. v_k \right)_i^* A_{ij}
\left. v_k \right)_j =  \mbox{LHS} \\
&&\mbox{Therefore,\ LHS\ is\ real}
\end{eqnarray*}

\item Formally, we can show this for
\begin{eqnarray*}
\langle x | A | y \rangle^* & = & \langle y | A^+ | x \rangle \\
& = & \langle y | A | x \rangle \mbox{if}\ A\ \mbox{Hermitean}
\end{eqnarray*}

\item Taking $\langle x | = \langle y |$, we see
$\langle x | A | x \rangle^* = \langle x | A | x \rangle$ if $A$
Hermitean

\item It follows that
\[ \lambda_k = \frac{\sum_{i,j=1}^n \left. v_k^* \right)_i A_{ij}
\left. v_k \right)_j}{\sum_{i=1}^n \left. v_k^* \right)_i \left. v_k 
\right)_i} \]
is ALWAYS real
\end{itemize}

\end{slide}

