Exercise 23, § 6

Let P be a real square matrix of type $(I,I)$; one says that P is stochastic if it is positive and if the vector $(1, . . . ,1)\in \mathbf{R}^I$ is an eigenvector of P associated with the eigenvalue 1.

Let A be a real square matrix of type $(I,I)$. Prove that A is positive and admits a strictly positive extremal vector if and only if there exists a stochastic matrix P, a real number $r >0$ and a diagonal matrix Z with strictly positive diagonal coefficients such that A $=r$ZPZ$^{-1}$. In this case the spectral radius of A is equal to $r$ and an extremal vector of A is Z$(1, . . . ,1)$.