We consider the two-sided eigenproblem <span class="tex">A * x = λ * B * x</span> over max algebra. It is shown that any finite system of real intervals and points can be represented as spectrum of this eigenproblem
In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero-diagonal real matrix A is characterized by A being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.