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On tropical Kleene star matrices and alcoved polytopes. (English) Zbl 1297.15029
A matrix \(A = (a_{ij}) \in {\mathbb R}^{n \times n}\) with zero diagonal induces the (possibly empty) alcoved polytope \[ C_A := \left\{ x \in {\mathbb R}^{n-1} : a_{in} \leq x_i \leq -a_{ni}, \, a_{ik} \leq x_i-x_k \leq -a_{ki}; \, i,k=1,\ldots,n-1; i \not= k \right\}. \] Then the main result of the paper states that a matrix \(A\) of the mentioned type is a Kleene star if and only if \[ C_A = \text{span}(A) \cap \left\{ x_n = 0 \right\}. \] The author gives an application of this to alcoved polytopes using normal idempotent matrices which form a subclass of Kleene stars: If \(A\) is a normal matrix, then \[ ||| A ||| = r (\text{span}(A) \cap \left\{ x_n=0 \right\}), \] where \(|||A||| := \text{max}_{i,j}|a_{ij}|\), and where \(r\) denotes the tropical radius of a subset of \({\mathbb R}^{n-1}\) containing the origin. If, in addition, \(A\) is idempotent then \(|||A||| = r(C_A)\).

MSC:
15A80 Max-plus and related algebras
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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