reciprocal polynomial
Definition [1]
Let
be a polynomial of degree with complex (or real)
coefficients. Then is a reciprocal polynomial if
for all .
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 is a zero for a reciprocal polynomial, then
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 ; if is an
eigenvalue
, so is .
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 |