15-00-CharacteristicPolynomialOfAOrthogonalMatrixIsAReciprocalPolynomial.tex

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\pmtitle{characteristic polynomial of a orthogonal matrix is a reciprocal polynomial}
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\begin{document}
\begin{thm}
The characteristic polynomial of a orthogonal matrix is a reciprocal polynomial
\end{thm}

\begin{proof}
Let $A$ be the orthogonal matrix, and let 
$p(\lambda) = \det(A-\lambda I)$ be 
its characteristic polynomial. We wish to prove that 
$$ 
  p(\lambda) = \pm \lambda^n p(1/\lambda).
$$
Since $A^{-1}=A^T$, we have 
$A-\lambda I=-\lambda A (A^T-I/\lambda ).$
Taking the determinant of both sides, and using 
$\det A = \det A^T$ and 
$\det c A = c^n \det A$ ($c\in \mathbb{C}$), 
yields
$$ \det (A-\lambda I) = \pm \lambda^n \det(A-\frac{1}{\lambda} I).$$
\end{proof}

\begin{thebibliography}{9}
\bibitem {eves} H. Eves,
 \emph{Elementary Matrix Theory}, Dover publications, 1980.
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