Matrix Completion and Decomposition in Phase Bounded Cones

TL;DR

This paper extends matrix completion and decomposition techniques from positive semidefinite cones to a broader class called phase-bounded cones, establishing duality and deriving conditions for solutions.

Contribution

It generalizes key results from PSD matrices to phase-bounded cones, introducing duality and characterizations for completion and decomposition problems.

Findings

Most PSD results carry over to phase-bounded cones.

Duality between completion and decomposition problems is established.

Characterization of all phase-bounded completions for banded matrices.

Abstract

The problem of matrix completion and decomposition in the cone of positive semidefinite (PSD) matrices is a well-understood problem, with many important applications in areas such as linear algebra, optimization, and control theory. This paper considers the completion and decomposition problems in a broader class of cones, namely phase-bounded cones. We show that most of the main results from the PSD case carry over to the phase-bounded case. More precisely, this is done by first unveiling a duality between the completion and decomposition problems, using a dual cone interpretation. Based on this, we then derive necessary and sufficient conditions for the phase-bounded completion and decomposition problems, and also characterize all phase-bounded completions of a completable partial matrix with a banded pattern.