Asymptotics of the Perron Eigenvalue and Eigenvector Using Max-Algebra

Marianne Akian (1) , Ravindra Bapat , Stéphane Gaubert (1)

1 META2 - Mathematical and Computer Science Tools for Automatic Control

We consider the asymptotics of the Perron eigenvalue and eigenvector of irreducible nonnegative matrices whose entries have a geometric dependance in a large parameter. The first term of the asymptotic expansion of these spectral elements is solution of a spectral problem in a semifield of jets, which generalizes the max-algebra. We state a «Perron-Frobenius theorem» in this semifield, which allows us to characterize the first term of this expansion in some non-singular cases. The general case involves an aggregation procedure à la Wentzell-Freidlin.

Keywords: PERRON-FROBENIUS THEOREM; MAX-ALGEBRA; PERTURBATION OF EIGENVALUES; PERTURBATION OF LINEAR OPERATORS; ASYMPTOTICS; FREIDLIN-WENTZELL THEORY; LARGE DEVIATIONS

Domains: Other [cs.OH]

Marianne Akian, Ravindra Bapat, Stéphane Gaubert. Asymptotics of the Perron Eigenvalue and Eigenvector Using Max-Algebra. [Research Report] RR-3450, INRIA. 1998. ⟨inria-00073240⟩