Solution Theory for Systems of Bilinear Equations

TL;DR

This paper develops a theoretical framework for analyzing the solvability of bilinear systems of equations, showing how solutions can be characterized through rank one completions and providing insights into conditions for universal solvability.

Contribution

It introduces a novel approach using rank one completions of linear matrix polynomials to analyze solvability of bilinear systems, including cases with many or few equations.

Findings

Solutions can be obtained as rank one completions of a derived linear matrix polynomial.

Identifies conditions under which bilinear systems are solvable for all right-hand sides.

Provides insights into the solvability of systems with varying numbers of equations.

Abstract

Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from elementary operations. This idea is used to identify bilinear systems that are solvable for all right hand sides and to understand solvability when the number of equations is large or small.