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Article Excerpt
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.
Key words: matrix rank-one decomposition; complex co-positivity cone; quadratic optimization; S-procedure
1. Introduction. The aim of this paper is to extend the results on the cone of nonnegative quadratic functions, studied by Sturm and Zhang in [9], from the real-valued to the complex domains. Sturm and Zhang [9] developed a matrix rank-one decomposition technique, which is a key technique in their approach to establish the linear matrix inequality representability of a class of matrix cones of nonnegative quadratic functions. It turns out that in the case of complex (Hermitian) quadratic forms, the rank-one decomposition result can actually be improved. In particular, we show in this paper that it is possible to find a rank-one decomposition for a positive semidefinite Hermitian matrix such that the inner-product between any of the rank-one matrices and two prescribed Hermitian matrices are constant, respectively. As a comparison, in the real case, the inner-product of these rank-one matrices and only one prescribed matrix can be made constant in general.
The organization of this paper is as follows. Section 2 studies such matrix rank-one decompositions, and [section] 3 is devoted to the description of the cone of nonnegative complex quadratic functions. The results of Sturm and Zhang [9] can be applied to solve quadratic optimization problems, as shown in Ye and Zhang [11]. Some of the results can be strengthened for the Hermitian forms, due to the results established in [section][section] 2 and 3. Section 4 is devoted to the complex quadratic programming problem. In [section] 5, we study the rank of optimal solutions for a standard complex SDR in light of the new rank-one decomposition result. Finally, in [section] 6 we investigate some interesting relationships between the rank-one decomposition theorem and the joint numerical range.
Notation. Throughout, we denote by [bar.a] the conjugate of a complex number a, and by [C.sup.n] the space of n-dimensional complex vectors. For a given vector z [member of] [C.sup.n], [z.sup.H] denotes the conjugate transpose of z. The space of n x n real symmetric and complex Hermitian matrices are denoted by [S.sup.n] and [H.sup.n], respectively. For a matrix Z [member of] [H.sup.n], we write Re Z and Im Z for the real and imaginary parts of Z, respectively. Matrix Z being Hermitian implies that Re Z is symmetric and Im Z is skew-symmetric. We denote by [S.sup.n.sub.+] ([S.sup.n.sub.++]) and [H.sup.n.sub.+]([H.sup.n.sub.++]) the cones of real symmetric positive semidefinite (positive definite) and complex Hermitian positive semidefinite (positive definite) matrices, respectively. The notation Z [greater than or equal to] (>0) means that Z is positive semidefinite (positive definite). For two complex matrices Y and Z, their inner product Y x Z = Re(tr [Y.sup.H]Z) = tr[[(Re Y).sup.T]r(Re Z) + [(Im Y).sup.T](Im Z)], where tr denotes the trace of a matrix and [sup.T] denotes the transpose of a matrix. For a square matrix M, diag(M) stands for a column vector whose elements are diagonal components of M.
2. A rank-one decomposition of Hermitian positive semidefinite (PSD) matrices. Let X [member of] [S.sup.n] be a real symmetric positive semidefinite matrix, and A [member of] [S.sup.n] be a real symmetric matrix. It follows by Sturm and Zhang [9] that there is a rank-one decomposition of X:
X = [r.summation over (j=1)][x.sub.j][x.sup.T.sub.j] such that [x.sup.T.sub.j]A[x.sub.j] = A x X/r, for j = 1, ..., r,
where r = rank X.
Now we shall show that in the Hermitian case the decomposition result can be extended to two matrices.
THEOREM 2.1. Suppose that Z [member of] [H.sup.n] is a complex Hermitian positive semidefinite matrix of rank r, and A, B [member of] [H.sup.n] be two given Hermitian matrices. Then, there is a rank-one decomposition of Z,
Z = [r.summation over (j=1)][z.sub.j][z.sup.H.sub.j],
such that
[z.sup.H.sub.j] A[z.sub.j] = A x Z/r, [z.sup.H.sub.j] B[z.sub.j] = B x Z/r, j = 1, ..., r.
PROOF. It follows from Sturm and Zhang [9, Corollary 4] that there is a decomposition of Z
Z = [r.summation over (j=1)][u.sub.j][u.sup.H.sub.j] such that [u.sup.H.sub.j]A[u.sub.j] = A...
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