Liénard-Chipart criterion

A modification of the Routh–Hurwitz criterion, which reduces all calculations in it to the calculation of the principal minors of only even (or only odd) orders of a Hurwitz matrix.

Suppose one is given a polynomial

(*)

let be its Hurwitz matrix (cf. Routh–Hurwitz criterion); let be its principal minor of order , .

The Liénard–Chipart criterion: Any of the following four conditions is necessary and sufficient in order that all roots of a polynomial (*) with real coefficients have negative real parts:

1) ;

2) ;

3) ;

4) .

The criterion was established by A. Liénard and H. Chipart [1].

References

[1]	A. Liénard, H. Chipart, "Sur la signe de la partie réelle des racines d'une équation algébrique" J. Math. Pures Appl. , 10 (1914) pp. 291–346
[2]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)