
\subsection{\bf The Definition of Scalar Products}
\subsection{\bf and Orthonormality}

\begin{slide}

\begin{itemize}
\item $\underline{x}^T {\fifteenbxti A} \underline{x}$ is a number (scalar)
written by physicists $\langle y \vert A \vert x \rangle$ where
$\langle y \vert$ is called a Bra.

\item The Bra-ket is $\langle y \vert x \rangle$ or scalar product

\item Therefore, 
\begin{eqnarray*}
A_{lk} & = & \underline{e}_l^T\, {\fifteenbxti A}\, \underline{e}_k \\
& = & \langle e_l \vert A \vert e_k \rangle
\end{eqnarray*}

The components of $A$ with respect to basis $\lbrace
\underline{e}_k \rbrace$

\item The basis $\lbrace e_k \rbrace$ is orthonormal
\begin{eqnarray*}
\langle \underline{e}_k \vert \underline{e}_l \rangle & = &
\delta_{lk} \\
& = & 0 \hspace{2em} l \neq k \\
& = & 1 \hspace{2em} l = k
\end{eqnarray*}

\item $\delta_{lk} = 0$ unless $l=k$ when $\delta_{lk} = 1$ is
Kronecker delta
\end{itemize}

\end{slide}

