LIANG Li1,WU Guoxing1*,CHEN Fei2, SHANG Shaoqiang1
(1 College of Science, Northeast Forestry University, Harbin 150040, Heilongjiang, China;2 College of Science, Beijing Information Science & Technology University, Beijing 100192, China)
Abstract:
The existence of the Hertime positive definite solution of matrix equation Xα+A*X-βA=I is investigated. The matrix equation has a solution X if and only if A admits the following factorization:A=(M*M)β2αN.By the CS decomposition theorem, the new necessary and sufficient conditions for the existence of the solution are obtained. The matrix equation has a solution if and only if there exist unitary matrices P and Q, and diagonal matrices C>0 and D≥0 with C2+D2=I such that A=P*CβαQDP.In this case, X=(P*C2P)1α is a solution; In the end, using the Brouwer fixed point theorem, if ‖A‖≤βα+ββααα+β, then equation has a solution x∈βα+βI,I.
KeyWords:
matrix equation; positive definite solution; the CS decomposition theorem; the Brouwer fixed point theorem
