I currently study control theory and read about positive linear systems. In Lorenzo Farina and Sergio Rinaldi's book: "Positive Linear Systems: Theory and Applications", chapter 5 about stability (theorem 13), the authors gave a criterion for asymptotic stability of the positive linear system :
A continuous-time $\dot{x}(t)=Ax(t)$ positive system is asymptotically stable if and only if the first $n$ leading minors of the matrix $-A$ are positive.
The authors said that this is simpler testing of Routh-Hurwitz's testing for a normally linear system, but I can not figure out how to deduce this testing from Hutwitz testing?
P/S: A continuous-time $\dot{x}(t)=Ax(t)$ system is positive if and only if A is Metzler matrix and asymptotically stable if and only if $A$ have eigenvalues with negative real parts. And for Routh–Hurwitz stability criterion: https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stability_criterion