摘要
研究了矩阵方程Xα+A*X-βA=I的Hermite正定解的存在性问题。首先,给出矩阵方程有解的充分必要条件,即存在一个Hermite正定阵M,使得矩阵A满足如下的分解:A=(M*M)β2αN;其次,在所得结论的基础上,利用CS分解定理,得到矩阵方程有解的另一个充分必要条件:存在酉矩阵P、Q以及对角矩阵C>0,D≥0,使得A=P*CβαQDP,其中C2+D2=I,CP=PC,此时方程的解可表示为X=(P*C2 P)1α;最后利用Brouwer不动点定理,证明若‖A‖≤βα+β+(αα+β)阵方程在区间[βα+βI,I]上有解X。
The existence of the Hertime positive definite solution of matrix equation Xα+A*X-βA=Iis investigated.The matrix equation has a solution Xif and only if Aadmits 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 Pand Q,and diagonal matrices C>0and D≥0with C2+D2=Isuch that A=P*CβαQDP.In this case,X=(P*C2 P)1αis a solution;In the end,using the Brouwer fixed point theorem,if‖A‖≤βα+β+(αα+β),then equation has a solution x∈[βα+β)I,I]
出处
《陕西师范大学学报（自然科学版）》
CAS
CSCD
北大核心
2015年第3期18-20,共3页
Journal of Shaanxi Normal University：Natural Science Edition
基金
国家自然科学基金资助项目(11401084)
关键词
矩阵方程
正定解
CS分解
BROUWER不动点定理
matrix equation
positive definite solution
the CS decomposition theorem
the Brouwer fixed point theorem
