Real Polynomial Gram Matrices Without Real Spectral Factors
Signal Processing
2019-03-12 v1
Abstract
It is well known that a non-negative definite polynomial matrix (a polynomial Gramian) can be written as a product of its polynomial spectral factors, . In this paper, we give a new algebraic characterization of spectral factors when is real-valued. The key idea is to construct a representation set that is in bijection with the set of real polynomial Gramians. We use the derived characterization to identify the set of all complex polynomial matrices that generate real-valued Gramians, and we formulate a conjecture that typical rank-deficient real polynomial Gramians have real spectral factors.
Cite
@article{arxiv.1903.04085,
title = {Real Polynomial Gram Matrices Without Real Spectral Factors},
author = {Puoya Tabaghi and Ivan Dokmanić},
journal= {arXiv preprint arXiv:1903.04085},
year = {2019}
}