  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.
