A note on the characteristic equation in the max-plus algebra

We discuss the characteristic equation of a matrix in the max-plus algebra. In their Linear Algebra Appl. paper [101:87–108 (1998)] Olsder and Roos have used a transformation between the max-plus algebra and linear algebra to show that the Cayley-Hamilton theorem also holds in the max-plus algebra. We show that the derivation of Olsder and Roos is not entirely correct, and we give the correct formulas for the coefficients of this alternative version of the max-algebraic characteristic equation. We also give a counterexample for a conjecture of Olsder in which he states necessary and sufficient for the coefficients of the max-algebraic characteristic equation.