I am reading about Positive Linear System in the book Positive Linear Systems Theory and Applications by LORENZO FARINA & SERGIO RINALDI. On chapter 5, page 40, I encounter Theorem 13, which the book doesn't include a proof:
A continuous-time $\dot{x}(t)=Ax(t)$ [discrete-time $x(t+1)=Ax(t)$] positive system is asymptotically stable if and only if the first n leading minors of the matrix $(-A)$ [$(I-A)$] are positive.
The discrete case can be obtained from the continuous case via a simple spectrum shift. I have found a proof for the continuous case in the book Positive 1D and 2D Systems by Tadeusz Kaczorek (Theorem 2.9, page 64).
For the "if" part, the idea is to use the positiveness of the $n$ leading minors of $-A$ to prove that all the principle minors are positive, making the coefficients of the characteristic polynomial positive. From here, the asymptotic stability of the system is obtained via Theorem 14 in the first book.
But in the proof of the second book, it simply state that the positivity of characteristic polynomial's coefficients is equivalent to the positivity of $n$ leading minors of $-A$.
My first question is about this statement. I don't understand how this can be done.
Another question is about my different idea of taking on the "only if" part: Does every $n-$dimension asymptotic stable positive linear system have a $(n-1)-$dimension asymptotic stable subsystem? If this is true then the "only if" part can be proved by the "if" part.
Any discussion is welcome. Thank you.