The phaseless rank of a matrix

Autor:	Goucha, António Pedro, Gouveia, João
Rok vydání:	2019
Předmět:	Mathematics - Algebraic Geometry Mathematics - Optimization and Control
Druh dokumentu:	Working Paper
Popis:	We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this quantity, extending a classic result of Camion and Hoffman and connecting it to the study of amoebas of determinantal varieties and of semidefinite representations of convex sets. As a consequence, we prove that the set of maximal minors of a matrix of indeterminates form an amoeba basis for the ideal they define, and we attain a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. We also highlight the connections between the notion of phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases. Comment: Changes in version 2: Some writing improvements, added some missing references and a section on the connection to equiangular lines
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1909.02417 Zobrazit plný text záznamu View this record from Arxiv