View source for Liénard-Chipart criterion

A modification of the [[Routh–Hurwitz criterion|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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587802.png" /> be its Hurwitz matrix (cf. [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587803.png" /> be its principal minor of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587805.png" />.

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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587806.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587807.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587808.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587809.png" />.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</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>