Let p′ and q′ be points in \Rn. Write p′∼q′ if p′−q′ is a multiple of (1,…,1). Two different points p and q in \Rn/∼ uniquely determine a tropical line L(p,q) passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on n leaves. It is also a metric graph. If some representatives p′ and q′ of p and q are the first and second columns of some real normal idempotent order n matrix A, we prove that the tree L(p,q) is described by a matrix F, easily obtained from A. We also prove that L(p,q) is caterpillar. We prove that every vertex in L(p,q) belongs to the tropical linear segment joining p and q. A vertex, denoted pq, closest (w.r.t tropical distance) to p exists in L(p,q). Same for q. The distances between pairs of adjacent vertices in L(p,q) and the distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the matrix |F|. In addition, if p and q are generic, then the tree L(p,q) is trivalent. The entries of F are differences (i. e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of A.
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.