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 $p(\lambda )=det(A-\lambda I)$ 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 $detA=det{A}^T$ 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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Title	characteristic polynomial of a orthogonal matrix is a reciprocal polynomial
Canonical name	CharacteristicPolynomialOfAOrthogonalMatrixIsAReciprocalPolynomial
Date of creation	2013-03-22 15:33:13
Last modified on	2013-03-22 15:33:13
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	5
Author	matte (1858)
Entry type	Theorem
Classification	msc 15-00
Related topic	CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial