In this paper, we present several new rank-one decomposition theorems for Hermitian positive semidefinite matrices, which generalize our previous results in Huang and Zhang (Math Oper Res 32(3):758-768, 2007), Ai and Zhang (SIAM J Optim 19(4):1735-1756, 2009). The new matrix rank-one decomposition theorems appear to have wide applications in theory as well as in practice. On the theoretical side, for example, we show how to further extend some of the classical results including a lemma due to Yuan (Math Program 47:53-63, 1990), the classical results on the convexity of the joint numerical ranges (Pang and Zhang in Unpublished Manuscript, 2004; Au-Yeung and Poon in Southeast Asian Bull Math 3:85-92, 1979), and the so-called Finsler's lemma (Bohnenblust in Unpublished Manuscript; Au-Yeung and Poon in Southeast Asian Bull Math 3:85-92, 1979). On the practical side, we show that the new results can be applied to solve two typical problems in signal processing and communication: one for radar code optimization and the other for robust beamforming. The new matrix decomposition theorems are proven by construction in this paper, and we demonstrate that the constructive procedures can be implemented efficiently, stably, and accurately. The URL of our Matlab programs is given in this paper. We strongly believe that the new decomposition procedures, as a means to solve non-convex quadratic optimization with a few quadratic constraints, are useful for many other potential engineering applications.
- Joint numerical range
- Matrix rank-one decomposition
- Positive semidefinite Hermitian matrix
- Quadratic optimization