reciprocal polynomial

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


for all z.

Examples of reciprocal polynomials are Gaussian polynomials, as well as the characteristic polynomialsMathworldPlanetmathPlanetmath of orthogonal matricesMathworldPlanetmath (including the identity matrixMathworldPlanetmath as a special case), symplectic matrices, involution matrices (, and the Pascal matricesMathworldPlanetmath [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 symmetricMathworldPlanetmathPlanetmath with respect to the unit circleMathworldPlanetmath in ; if λ is an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath, so is 1/λ.

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


  • 1 H. Eves, Elementary MatrixMathworldPlanetmath 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