reciprocal polynomial

Definition [1] Let $p:ℂ\to ℂ$ be a polynomial of degree $n$ with complex (or real) coefficients. Then $p$ is a reciprocal polynomial if

$$p(z)=\pm z^{n}p(1/z)$$

for all $z\in ℂ$.

Examples of reciprocal polynomials are Gaussian polynomials, as well as the characteristic polynomials of orthogonal matrices (including the identity matrix as a special case), symplectic matrices, involution matrices (http://planetmath.org/LinearInvolution), and the Pascal matrices [2].

It is clear that if $z$ is a zero for a reciprocal polynomial, then $1/z$ is also a zero. This property motivates the name. This means that the spectra of matrices of above type is symmetric with respect to the unit circle in $ℂ$; if $\lambda \in ℂ$ is an eigenvalue, so is $1/\lambda$.

The sum, difference, and product of two reciprocal polynomials is again a reciprocal polynomial. Hence, reciprocal polynomials form an algebra over the complex numbers.

Title	reciprocal polynomial
Canonical name	ReciprocalPolynomial
Date of creation	2013-03-22 13:36:33
Last modified on	2013-03-22 13:36:33
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	12
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 12D10
Related topic	CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial