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Abstract
The problem of finding a rankone solution to a system of linear matrix equations arises from many practical applications. Given a system of linear matrix equations, however, such a lowrank solution does not always exist. In this paper, we aim at developing some sufficient conditions for the existence of a rankone solution to the system of homogeneous linear matrix equations (HLME) over the positive semidefinite cone. First, we prove that an existence condition of a rankone solution can be established by a homotopy invariance theorem. The derived condition is closely related to the socalled P∅ property of the function defined by quadratic transformations. Second, we prove that the existence condition for a rankone 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 rankone solution to the system of HLME is also established in this paper.
Original language  English 

Pages (fromto)  55695583 
Number of pages  14 
Journal  Applied Mathematics and Computation 
Volume  219 
DOIs  
Publication status  Published  2013 
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Dive into the research topics of 'Rankone solutions for homogeneous linear matrix equations over the positive semidefinite cone'. Together they form a unique fingerprint.Projects
 1 Finished

Foundation and Reweighted Algorithms for Sparsest Points of Convex Sets with Application to Data Processing
Zhao, Y.
Engineering & Physical Science Research Council
18/04/13 → 31/05/15
Project: Research Councils