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In this paper, we introduce and study the notion of phase rank. Given a matrix filled with phases, i.e., with complex entries of modulus $1$, its phase rank is the smallest possible rank for a complex matrix with the same phase, but possible different modulus. This notion generalizes the notion of sign rank, and is the complementary notion to that of phaseless rank. It also has intimate connections to the problem of ray nonsingularity and provides an important class of examples for the study of coamoebas. Among other results, we provide a new characterization of phase rank $2$ for $3 \times 3$ matrices, that implies the new result that for the $3\times 3$ determinantal variety, its coamoeba is determined solely by the colopsidedness criterion.

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

In this paper we study various versions of extension complexity for polygons through the study of factorization ranks of their slack matrices. In particular, we develop a new asymptotic lower bound for their nonnegative rank, shortening the gap between the current bounds, we introduce a new upper bound for their boolean rank, deriving from it some new numerical results, and we study their complex semidefinite rank, uncovering the possibility of non monotonicity of the ranks of regular $n$-gons., Comment: [Updated funding info and corrected typos]

In this paper the introduction of notion of reference vector paves the way for a combination of classical and social approaches in the framework of referential preferences given by matrix groups. It is shown that individual demand issue from rational decision does not depend on that reference.