Exercise 24, § 6

(a) Let $P$ be a real stochastic square matrix of order $n$. The real number 1 is a simple root of the minimal polynomial of $P$. (One may use Exercise 20.) Moreover, every root of modulus 1 of the minimal polynomial of $P$ is simple.

(b) Let $A$ be a positive real matrix admitting an extremal vector $\succ 0$ and of spectral radius $\varrho$. All the roots of modulus $\varrho$ of the minimal polynomial of $A$ are simple.

In Exercises 25 to 30, one considers a real Fréchet space E and a closed convex cone C of E, supposed to have nonempty interior. The order relation on E induced by C is denoted by $\preccurlyeq_C$. Thus, for $x,y$ in E, one has $x\succcurlyeq_Cy$ if and only if $x-y\in C$. One shall write $x\succ_Cy$ for the relation $x-y\in \mathring{C}$.

One considers a continuous mapping $u: E\rightarrow E$ satisfying the following conditions:

(i) The mapping $u$ is positively homogeneous: one has $u(\lambda x) =\lambda u(x)$ for every real number $\lambda \geqslant 0$ ;

(ii) The mapping $u$ is increasing with respect to C, that is to say that $x\succcurlyeq_Cy$ implies $u(x)\succcurlyeq_Cu(y)$;

(iii) For every $x\in C-\{0\}$, one has $u(x)\in \mathring{C}$.

The purpose of these exercises, taken from the course of P.-L. Lions at the Collège de France, 2020/2021, is to prove, under certain conditions, that there exist an element $x\in C-\{0\}$ and $\lambda >0$ such that $x=\lambda u(x)$; this provides in particular another proof of the Krein–Rutman theorem (III, p. 94, th. 4 and III, p. 97, prop. 8).

For $f\in \mathring{C}$ and $\lambda \in \mathbf{R}_+$, one calls an upper solution (resp. lower solution) of the equation $y=\lambda u(y)+f$ an element $y\in C$ such that $y\succcurlyeq_C\lambda u(y)+f$ (resp. such that $y\preccurlyeq_C\lambda u(y) +f$). One denotes by $U_f$ the set of the $\lambda \in \mathbf{R}_+$ such that the set of upper solutions of the equation $y=\lambda u(y) +f$ is nonempty.