This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang in 2003. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of LMI. We also present an upper bound on the minimum rank among optimal solutions for a standard complex SDP problem, as a byproduct of the new rank-one decomposition result.
Technical Report SEEM2005-02, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, 2005