A Hurwitz polynomial is a polynomial with real positive coefficients and roots which are either negative or occur in complex conjugate pairs with negative real parts.
The positive-coefficient normalization fixes an overall sign. Gantmacher (1964, p. 195) calls any stable polynomial a Hurwitz polynomial, regardless
of the sign of its leading coefficient. For
a positive leading
coefficient, positivity of the other coefficients
follows by polynomial factorization over
the real numbers. Apart from the positive leading coefficient, the factors
have the forms 
and 
, where 
and 
is real. Each factor
has positive coefficients.
Dividing any stable polynomial by its leading
coefficient therefore produces a monic polynomial
that is a Hurwitz polynomial.
Positive coefficients alone are not sufficient. For example,
 |
The roots 
have positive real
part. The Routh-Hurwitz theorem gives
a criterion in terms of the coefficients.
