The phase rank of a matrix

TL;DR

This paper introduces the concept of phase rank for matrices with complex entries of modulus one, generalizing sign rank and connecting to coamoebas, with new characterizations for specific cases.

Contribution

It defines phase rank, explores its properties, and provides a new characterization for phase rank 2 in 3x3 matrices, linking to coamoeba structures.

Findings

Phase rank generalizes sign rank.

Characterization of phase rank 2 for 3x3 matrices.

Coamoeba of the 3x3 determinantal variety is determined by colopsidedness.

Abstract

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.