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''Hurwitz criterion''

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

$$f(x)=a_0x^n+a_1x^{n-1}+\dotsb+a_n,$$

with real coefficients and $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\Delta_i$, $i=1,\dotsc,n$, of the Hurwitz matrix $H$ are positive (cf. [[Minor|Minor]]). Here $H$ is the matrix of order $n$ whose $i$-th row has the form

$$a_{2-i},a_{4-i},\dotsc,a_{2n-i},$$

where, by definition, $a_k=0$ if $k<0$ or $k>n$ (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz [[#References|[1]]] and is a generalization of the work of E.J. Routh (see [[Routh theorem|Routh theorem]]).

A polynomial $f(x)$ 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|Liénard–Chipart criterion]], and methods for determining the number of real roots of a polynomial are also known.

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



====Comments====
See also [[Routh theorem|Routh theorem]].