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\title{\bf A Geometric Interpretation of the Principal Invariants  %@@
 of a Real Symmetric Matrix}  %@@
\author{\large Roberto B. Salgado}  %@@
\maketitle  %@@
\abstract{We present a geometric interpretation  %@@
of the principal invariants of a real symmetric $n$-by-$n$ matrix.  %@@
We will show that the $k^{th}$ principal invariant (where $1 \leq k \leq n$)  %@@
is a measure of the bounding $k$-volume of an $n$-dimensional rectangular  %@@
parallelopiped.  %@@



\section{Introduction}  %@@

The $k^{th}$ principal invariant $I_k=A^{[a}{}_{a}\cdots A^{k]}{}_{k}$  %@@
is the coefficient of the $\lambda^{n-k+1}$ term in the characteristic  %@@
polynomial of $A^a{}_{b}$.  It is equal to the $k^{th}$ elementary symmetric  %@@
function of the eigenvalues of $A^a{}_{b}$.  %@@



Let $Q(k,n)$ denote the number of $k$-cubes in an $n$-cube.  %@@

We need the number of $k$-cubes at each vertex.  %@@

We need the number if edges at each vertex: this number is $n$.  %@@

To form a $k$-cube, one chooses $k$ distinct edges from those $n$ edges:  %@@
   the number of ways to do this is $\frac{n!}{k!\ (n-k)!}$.  %@@

The number of vertices in an $n$ cube is $2^n$.  %@@

Hence, $$Q(k,n)=\frac{(2^n)\left(\frac{n!}{k!\ (n-k)!}\right)}{2^k},$$  %@@
since each $k$-cube is counted $2^k$ times.  %@@

Hence, the $k^{th}$-invariant $$I_k=\frac{1}{2^{n-k}} \tilde Q(k,n),$$  %@@
where $\tilde Q(k,n)$ is the $k$-volume-weighted number of $k$-cubes  %@@
in this $n$-prism.  %@@



\begin{thebibliography}{9}  %@@

\bibitem{crc} {\it CRC Standard Mathematical Tables}, 26th-ed  %@@
(Boca Raton, Florida, CRC Press.) 1981.  %@@

\bibitem{mtw}  %@@
Misner, C. W. / Thorne, K. / Wheeler, J. A.  %@@
{\it Gravitation}  %@@

\bibitem{burke-adg}  %@@
Burke, W. L.,  %@@
{\it Applied Differential Geometry},  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{burke-sgc}  %@@
Burke, W. L.,  %@@
{\it Spacetime, Geometry and Cosmology},  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{schouten-ricci}  %@@
Schouten, J. A.,  %@@
{\it Ricci-Calculus},  %@@
(Berlin, Springer-Verlag) 1954.  %@@

\bibitem{schouten-tensor}  %@@
Schouten, J. A.,  %@@
{\it Tensor Analysis for Physicists},  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{bamberg-sternberg}  %@@
Bamberg, P. / Sternberg, S.  %@@
{\it A course in mathematics for students of physics},  %@@
(Cambdrige, Cambridge University Press) 1990.  %@@

\bibitem{vandanzig}  %@@
van Danzig, ?,  %@@
{\it ? },  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{vnr}Gellert / K\"ustner / Hellwich / K\"astner,  %@@
{\it The VNR Concise Encyclopedia of Mathematics}  %@@
(New York, Van Nostrand Reinhold) 1975.  %@@

\end{thebibliography}  %@@

\ni{\em Department of Physics\\  %@@
        Syracuse University\\  %@@
        Syracuse NY 13244-1130\\  %@@
   {\tt salgado@suhep.phy.syr.edu}  %@@
   }  %@@

\end{document}  %@@
\end  %@@

