reciprocal polynomial
Definition [1] Let $p:\u2102\to \u2102$ 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 $z\in \u2102$.
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 $\u2102$; if $\lambda \in \u2102$ 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^{}.
References
- 1 H. Eves, Elementary Matrix^{} Theory, Dover publications, 1980.
- 2 N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.
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 |