Publication: Tropical linear maps on the plane
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2010-09-22
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Elsevier
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Abstract
In this paper we fully describe all tropical linear maps in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by a real 3×3 matrix A. The map fA is continuous and piecewise-linear in the classical sense. In some particular cases, the map fA is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the map collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (Theorem 3).
In order to study fA, we may assume that A is normal, i.e., I A 0, up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique normalization for A (which we call lower canonical normalization) (Theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning.
On , any , some aspects of tropical linear maps have been studied in [6]. We work in , adding a geometric view and doing everything explicitly. We give precise pictures.
Inspiration for this paper comes from [3,5,6,8,12,26]. We have tried to make it self-contained. Our preparatory results present noticeable relationships between the algebraic properties of a given matrix A (idempotent normal matrix, permutation matrix, etc.) and classical geometric properties of the points spanned by the columns of A (classical convexity and others); see Theorem 2 and Corollary 1. As a by-product, we compute all the tropical square roots of normal matrices of a certain type; see Corollary 4. This is, perhaps, a curious result in tropical algebra. Our final aim is, however, to give a precise description of the map . This is particularly easy when two tropical triangles arising from A (denoted and ) fit as much as possible. Then the action of fA is easily described on (the closure of) each cell of the cell decomposition ; see Theorem 3.
Normal matrices play a crucial role in this paper. The tropical powers of normal matrices of size satisfy A n-1=A n=A n+1= . This statement can be traced back, at least, to [26], and appears later many times, such as [1,2,6,9,10]. In lemma 1, we give a direct proof of this fact, for n=3. But now the equality A 2=A 3 means that the columns of A 2 are three fixed points of fA and, in fact, any point spanned by the columns of A 2 is fixed by fA. Among 3×3 normal matrices, the idempotent ones (i.e., those satisfyingA=A 2) are particularly nice: we prove that the columns of such a matrix tropically span a set which is classically compact, connected and convex (Lemma 2 and Corollary 1). In our terminology, it is a good tropical triangle