Complex matrix decomposition and quadratic programming

This paper studies the possibilities of the linear matrix inequality 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 [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246-267]. 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 [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246-267] 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 linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform. Mathematische Zeitschrift 3 314-316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix. Proc. Amer. Math. Soc. 12 61-66] on the joint numerical range.