Title: 
New Results on Hermitian Matrix Rank-One Decomposition

Authors:
Wenbao Ai
School of Science
Beijing University of Posts and Telecommunications
People's Republic of China
wenbaoai@gmail.com

Yongwei Huang
Department of Systems Engineering and Engineering Management
The Chinese University of Hong Kong
Shatin, Hong Kong
ywhuang@se.cuhk.edu.hk

Shuzhong Zhang
Department of Systems Engineering and Engineering Management
The Chinese University of Hong Kong
Shatin, Hong Kong
zhang@se.cuhk.edu.hk


Abstract:
In this paper, we present several new rank-one decomposition theorems for Hermitian positive semidefinite matrices,
which generalize our previous results~\cite{h-z05,A_Z06}.The new matrix rank-one decomposition theorems have wide applications in theory and in practice. On the theoretical side, as examples, we show how
to further extend some of the classical results based on our new results, including a lemma due to
Yuan~\cite{yuan90mp}, and some classical results on the convexity of the joint numerical ranges~\cite{PZ04, AYP79}, and Finsler's
lemma~\cite{bohn, AYP79}. The matrix decomposition theorems that we developed are constructive and can be
implemented stably and accurately. The URL of our Matlab codes is given in this paper. We believe that the new decomposition procedures are useful for many potential engineering applications.