In mathematics, the Routh–Hurwitz matrix,[1] or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the polynomial.
It was established by Adolf Hurwitz in 1895 that a real polynomial with
is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix
are called the Hurwitz determinants.
then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.
As an example, consider the matrix and let be the characteristic polynomial of
is then The leading principal minors of
are Since the leading principal minors are all positive, all of the roots of
have negative real part.
have negative real part, and hence