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The problem of finding a rank-one solution to a system of linear matrixequations arises from many practical applications. Given a system of linearmatrix equations, however, such a low-rank solution does not always exist. Inthis paper, we aim at developing some sufficient conditions for the existenceof a rank-one solution to the system of homogeneous linear matrix equations(HLME) over the positive semidefinite cone. First, we prove that an existencecondition of a rank-one solution can be established by a homotopy invariancetheorem. 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 bealso established through the maximum rank of the (positive semidefinite) linearcombination of given matrices. It is shown that an upper bound for the rank ofthe solution to a system of HLME over the positive semidefinite cone can beobtained efficiently by solving a semidefinite programming (SDP) problem.Moreover, a sufficient condition for the nonexistence of a rank-one solution tothe system of HLME is also established in this paper.

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