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Three notions of tropical rank for symmetric matrices

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Abstract

We introduce and study three different notions of tropical rank for symmetric 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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Authors and Affiliations

  1. Department of Mathematics, Yale University, P.O. Box 208283, New Haven, CT, 06520-8283, USA

    Dustin Cartwright

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Melody Chan

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  1. Dustin Cartwright
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  2. Melody Chan
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Correspondence to Dustin Cartwright.

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Cartwright, D., Chan, M. Three notions of tropical rank for symmetric matrices. Combinatorica 32, 55–84 (2012). https://doi.org/10.1007/s00493-012-2701-4

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  • DOI: https://doi.org/10.1007/s00493-012-2701-4

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