Séminaire de Géométrie Tropicale
Mercredi 14 janvier, 16h15 salle 1525-502
Adi Niv
(INRIA Saclay)
Supertropical matrix algebra
Résumé :
The Max-Plus (tropical) algebra, is the set of real numbers, together with $-\infty$, equipped with the operations maximum and the usual plus. We start by presenting some basic notation in this setting, and show how the lack of additive inverse causes failure of some classic algebraic properties. Next, we present the extended (supertropical) algebra, which adds a layer of singular elements to R, and show how this extension offers a solution to these failed properties. In the second part we introduce definitions and theorems in supertropical linear algebra, and state the connection between the eigenvalues of a matrix to those of its powers, tropical quasi-inverse and tropical conjugates. We introduce the definite matrices as a key tool for proving these results. If time allows, we will present miscellaneous results on factorization of non-singular matrices into a product of tropical elementary matrices.
The results in this talk are a part of the speaker's PhD thesis.