On factorization of rank-one auto-correlation matrix polynomials

TL;DR

This paper provides a comprehensive analysis of the conditions under which rank-one auto-correlation matrix polynomials can be uniquely factored, offering explicit methods for factorization and enumeration of solutions in non-unique cases.

Contribution

It establishes a necessary and sufficient condition for the uniqueness of factorization based on the GCD of polynomials and details explicit factorization methods.

Findings

Unique factorization condition based on GCD established

Explicit factorization method provided for the unique case

Enumeration of all solutions in non-unique cases

Abstract

This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of multiple polynomials. In the unique case, we show that the factorization can be carried out explicitly using GCDs. In the non-unique case, the number of non-trivially different factorizations is given and all solutions are enumerated.