# characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

###### Proof.

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

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

## References

Title characteristic polynomial of a orthogonal matrix is a reciprocal polynomial CharacteristicPolynomialOfAOrthogonalMatrixIsAReciprocalPolynomial 2013-03-22 15:33:13 2013-03-22 15:33:13 matte (1858) matte (1858) 5 matte (1858) Theorem msc 15-00 CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial