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 $detcA={c}^{n}detA$ ($c\in \u2102$), yields
$$det(A-\lambda I)=\pm {\lambda}^{n}det(A-\frac{1}{\lambda}I).$$ |
∎
References
- 1 H. Eves, Elementary Matrix^{} Theory, Dover publications, 1980.
Title | characteristic polynomial of a orthogonal matrix is a reciprocal polynomial |
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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 |