# reciprocal polynomial

Definition [1] Let $p:\mathbb{C}\to\mathbb{C}$ 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 $z\in\mathbb{C}$.

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 $\mathbb{C}$; if $\lambda\in\mathbb{C}$ 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.

## 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 ReciprocalPolynomial 2013-03-22 13:36:33 2013-03-22 13:36:33 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Definition msc 12D10 CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial