Hurwitz determinant

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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.


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 i-th Hurwitz determinant is the i-th leading principal minor (minor is a determinant) of the above Hurwitz matrix H. There are n Hurwitz determinants for a characteristic polynomial of degree n.

See also[edit]


  • Hurwitz, A. (1895), "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt", Mathematische Annalen, 46 (2): 273–284, doi:10.1007/BF01446812, S2CID 121036103
  • Wall, H. S. (1945), "Polynomials whose zeros have negative real parts", The American Mathematical Monthly, 52 (6): 308–322, doi:10.1080/00029890.1945.11991574, ISSN 0002-9890, JSTOR 2305291, MR 0012709