As P is an irreducible, Perron-Frobénius theorem (see [3] for example) assure that λ is real and the sub-stochastic property, that λ ≤ 1 (in the particular case where P is also primitive i.e. ∃m, Pm > 0 then |λ2| <λ) Replacing x by Iℓ equation (51) gives the expression of the row ℓ of Pi and the proposition is then proved.     □