Charact​eristic Polynomials of Supertropical Matrices

The Max-Plus (tropical) algebra, is the set of real numbers R, 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. Then, we present the extended (supertropical) algebra, introduced and studied by Izhakian and Rowen, which adds a layer of singular elements to R. We show how this extension recovers these failed properties. In the last 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-inverse and conjugates. If time allows, we will give some details of the proof.

 

*The results on characteristic polynomials are a part of the speaker's PhD thesis.