133 Necessary condition for stability

Suppose that $h(s)$ is a monic real polynomial that is stable. Then all the coefficients of $h(s)$ are positive.

Proof. The roots of $h(s)$ are either real ; or pairs of conjugate complex roots ${\mu }_k$ and ${\overline{\mu }}_k$ with , which combine to give real quadratic factors. Hence $h(s)$ factorizes as

where $-{\lambda }_j>0,-2\mathrm{\Re }{\mu }_k>0$ and ${|{\mu }_k|}^2>0$; hence all the coefficients we obtain on multiplying out are positive.