Routh-Hurwitz criterion

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Hurwitz criterion

A necessary and sufficient condition for all the roots of a polynomial

with real coefficients and , to have negative real parts. It consists of the following: All principal minors , , of the Hurwitz matrix are positive (cf. Minor). Here is the matrix of order whose -th row has the form

where, by definition, if or (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz [1] and is a generalization of the work of E.J. Routh (see Routh theorem).

A polynomial satisfying the Hurwitz condition is called a Hurwitz polynomial, or, in applications of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynomials, such as the Routh criterion, the Liénard–Chipart criterion, and methods for determining the number of real roots of a polynomial are also known.


[1] A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt" Math. Ann. , 46 (1895) pp. 273–284
[2] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)


See also Routh theorem.

How to Cite This Entry:
Routh-Hurwitz criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article