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# 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, 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.

## Mathematics Subject Classification

12D10*no label found*

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