Rank-one Solutions for Homogeneous Linear Matrix Equations over the Positive Semidefinite Cone

TL;DR

This paper investigates conditions under which a rank-one solution exists for homogeneous linear matrix equations over the positive semidefinite cone, providing theoretical criteria and efficient computational methods.

Contribution

It introduces new sufficient conditions for the existence of rank-one solutions, linking homotopy invariance and quadratic transformation properties, and offers an SDP-based approach to bound solution ranks.

Findings

Existence condition via homotopy invariance theorem.

Bound on solution rank through SDP optimization.

Sufficient condition for nonexistence of rank-one solutions.

Abstract

The problem of finding a rank-one solution to a system of linear matrix equations arises from many practical applications. Given a system of linear matrix equations, however, such a low-rank solution does not always exist. In this paper, we aim at developing some sufficient conditions for the existence of a rank-one solution to the system of homogeneous linear matrix equations (HLME) over the positive semidefinite cone. First, we prove that an existence condition of a rank-one solution can be established by a homotopy invariance theorem. The derived condition is closely related to the so-called $P_\emptyset$ property of the function defined by quadratic transformations. Second, we prove that the existence condition for a rank-one solution can be also established through the maximum rank of the (positive semidefinite) linear combination of given matrices. It is shown that an upper bound for the rank of the solution to a system of HLME over the positive semidefinite cone can be obtained efficiently by solving a semidefinite programming (SDP) problem. Moreover, a sufficient condition for the nonexistence of a rank-one solution to the system of HLME is also established in this paper.