

\subsection{\bf Errors and Moment Matrices}

\begin{slide}
\begin{description}
\item{$\bullet$} The Moment Matrix is used to quantify errors in
experimental measurements.

\begin{itemize}
\item Often, the results of an experiments on $x_1 \ldots x_n$ are
quoted in terms of means $\pm$ errors:

\item These errors are standard deviations $= \sqrt{\mbox{diagonal
elements of moment matrix}}$.

\item Now, this is throwing away information---the off diagonal terms
in $M$---which can be important.

\item Thus, suppose some years after the experiment measures and
reports $\Underline{$x$}$, a theorist decides that particular linear
combinations $\Underline{$y$} = T\Underline{$x$}$ ($T$ an $n\times n$
matrix) are useful.

\item What are errors on $y$?
\begin{eqnarray}
\left\langle y_k y_l\right\rangle & = & T_{kk^\prime} \left\langle x_{k^\prime}
x_{l^\prime} \right\rangle T^T_{l^\prime l} \hspace{2em} \mbox{or} \nonumber \\
M_y & = & TM_x T^T
\end{eqnarray}

\item To find diagonal terms in $M_y$---i.e., the errors$^2$ on our new
theorist's $y$---requires off diagonal terms in $M_x$.
\end{itemize}
\end{description}
\end{slide}

