characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

Theorem 1.

The characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

Proof.

Let $A$ be the orthogonal matrix, and let$p(\lambda )=det(A-\lambda I)$ beits 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$detA=det{A}^T$ and$detcA=c^{n}detA$ ($c\in ℂ$),yields

$$det(A-\lambda I)=\pm {\lambda }^{n}det(A-\frac{1}{\lambda }I).$$

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