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Three notions of tropical rank for symmetric matrices

Abstract : We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull.
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Dustin Cartwright, Melody Chan. Three notions of tropical rank for symmetric matrices. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.203-214, ⟨10.46298/dmtcs.2865⟩. ⟨hal-01186294⟩



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