In mathematics, Hurwitz determinants were introduced by Adolf Hurwitz (1895), who used them to give a criterion for all roots of a polynomial to have negative real part.
Let us consider a characteristic polynomial P in the variable λ of the form:
where , , are real.
The square Hurwitz matrix associated to P is given below:
The ith Hurwitz determinant is the determinant of the ith leading principal minor of the above Hurwitz matrix H. There are n Hurwitz determinants for a characteristic polynomial of degree n.
- Hurwitz, A. (1895), "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt", Mathematische Annalen, 46 (2), doi:10.1007/BF01446812
- Wall, H. S. (1945), "Polynomials whose zeros have negative real parts", The American Mathematical Monthly, 52: 308–322, ISSN 0002-9890, JSTOR 2305291, MR 0012709