Real Polynomial Gram Matrices Without Real Spectral Factors

TL;DR

This paper provides a new algebraic characterization of spectral factors for real polynomial Gram matrices, identifying conditions under which these matrices have real spectral factors and exploring their structure.

Contribution

It introduces a novel algebraic framework to characterize spectral factors of real polynomial Gram matrices and identifies conditions for the existence of real spectral factors.

Findings

Characterization of spectral factors for real polynomial Gramians

Identification of complex polynomial matrices generating real Gramians

Conjecture on real spectral factors for rank-deficient Gramians

Abstract

It is well known that a non-negative definite polynomial matrix (a polynomial Gramian) $G(t)$ can be written as a product of its polynomial spectral factors, $G(t) = X(t)^H X(t)$. In this paper, we give a new algebraic characterization of spectral factors when $G(t)$ 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.