Definite Determinantal Representations via Orthostochastic Matrices
Abstract
Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a determinantal representation. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree . In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. We also characterize the range set of vector coefficients of a certain type of determinantal bivariate polynomials.
Cite
@article{arxiv.1708.09559,
title = {Definite Determinantal Representations via Orthostochastic Matrices},
author = {Papri Dey},
journal= {arXiv preprint arXiv:1708.09559},
year = {2019}
}