Eigenvalues and Eigenfunctionals of Diagonally Dominant Endomorphisms in Min-Max Analysis - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Linear Algebra and its Applications Année : 1998

Eigenvalues and Eigenfunctionals of Diagonally Dominant Endomorphisms in Min-Max Analysis

Michel Gondran
  • Fonction : Auteur
Michel Minoux
  • Fonction : Auteur
  • PersonId : 846863

Résumé

The so-called (Min, +) analysis may be viewed as an extension to the continuous case and to functional spaces of shortest path algebras in graphs. We investigate here (Min-Max) analysis which extends, in some similar way, minimum spanning tree problems and maximum capacity path problems in graphs. An endomorphisms $A$ of the functional Min-Max semi-module acts on any functional $f$ to produce $Af$, where, $\forall x$: $$Af(x) = \min_{y}\{\max\{A(x,y);f(y)\}\}.$$ We present here a complete characterization of eigenvalues and eigen-functionals of diagonally dominant endomorphisms (i.e. such that $\forall x$, $\forall y$: $A(x, x) = 0_A$, $A(x, y) \geq 0_A$). It is shown, in particular, that any real value $\lambda > 0_A$ is an eigenvalue, and that the associated eigen-semi-module has a unique minimal generator.

Dates et versions

hal-01150536 , version 1 (11-05-2015)

Identifiants

Citer

Michel Gondran, Michel Minoux. Eigenvalues and Eigenfunctionals of Diagonally Dominant Endomorphisms in Min-Max Analysis. Linear Algebra and its Applications, 1998, 282 (1-3), pp.47-61. ⟨10.1016/S0024-3795(98)10039-3⟩. ⟨hal-01150536⟩
42 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More