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Tropicalizing the positive semidefinite cone


Author: Josephine Yu
Journal: Proc. Amer. Math. Soc. 143 (2015), 1891-1895
MSC (2010): Primary 14T05, 52B12, 26A51
Published electronically: November 24, 2014
MathSciNet review: 3314099
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Abstract: We study the tropicalization of the cone of positive semidefinite matrices over the ordered field of real Puiseux series. The tropical PSD matrices form the normal cone of the Newton polytope of the symmetric determinant at the vertex corresponding to the product of diagonal entries. We find generators and defining inequalities of the cone. The PSD tropical quadratic forms are those that induce the trivial subdivision on the standard simplex dilated by two. We also show that the tropical PSD cone is the tropical convex hull of the set of symmetric matrices of tropical rank one and that every tropical PSD matrix can be factored as a tropical product of a matrix and its transpose.


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Additional Information

Josephine Yu
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: jyu@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12428-2
Received by editor(s): September 26, 2013
Published electronically: November 24, 2014
Additional Notes: The author was supported by the NSF-DMS grant #1101289
Communicated by: Jim Haglund
Article copyright: © Copyright 2014 American Mathematical Society