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reciprocal polynomial
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(Definition)
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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, 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.
- 1
- H. Eves, Elementary Matrix Theory, Dover publications, 1980.
- 2
- N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.
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"reciprocal polynomial" is owned by rspuzio. [ full author list (3) | owner history (2) ]
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Cross-references: complex numbers, algebra, product, difference, sum, eigenvalue, unit circle, symmetric, type, matrices, property, clear, Pascal matrices, symplectic matrices, identity matrix, orthogonal matrices, characteristic polynomials, Gaussian polynomials, coefficients, real, complex, polynomial
There are 3 references to this entry.
This is version 9 of reciprocal polynomial, born on 2003-05-03, modified 2006-10-17.
Object id is 4239, canonical name is ReciprocalPolynomial.
Accessed 3511 times total.
Classification:
| AMS MSC: | 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros ) |
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Pending Errata and Addenda
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