Eigenvalues and Eigenvectors of Endomorphisms

The celebrated Perron-Frobenius theorem, which applies to real nonnegative matrices, may be viewed as the first result stating the existence of an eigenvalue and associated eigenvector on matrices with coefficients in the dioid (R₊,+,×). Indeed, it asserts that such a matrix has an eigenvalue in this dioid, with an associated eigenvector having all components in the dioid; moreover, it establishes a special property for this eigenvalue, as compared with the other eigenvalues on the field of complex numbers: it is actually the one having the largest modulus.

The importance of this largest eigenvalue is well-known as it is often related to stability issues for dynamical systems (Lyapounov coefficient), or to asymptotic behavior of systems (see, e.g. Exercise 2 at the end of this chapter).

The present chapter is devoted to the characterization of eigenvalues and eigenvectors for endomorphisms of semi-modules and of moduloids in finite dimensions. Extension to functional operators in infinite dimensions will be studied in Exercise 3 of this chapter (for Max+ dioids) and in Chap. 7, Sect. 4 (for Min–Max dioids).