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, 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.

Bibliography

1

H. Eves, Elementary Matrix Theory, Dover publications, 1980.

2

N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.