关于非线性矩阵方程X~s+A~*X~(-t)A+B~*X~(-t)B=Q的若干结果
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摘要
主要研究非线性矩阵方程X~s+A~*X~(-t)A+B~*X~(-t)B=Q的正定解,其中A、B为n阶复矩阵,s,t为正整数,Q为n阶正定矩阵.首先,基于线性方程组理论,研究了该矩阵方程正定解的性质.其次,在特殊条件下,研究了该矩阵方程解的存在区间.
In this paper,the Hermitian positive definite solution of the nonlinear matrix equation X~s+A~*X~(-t)A+B~*X~(-t)B=Q is investigated,in which A、Bare n×ncomplex matrices,s,t are positive integers and Qis a positive definite matrix.Firstly,based on linear system of equations theory,the property of the solutions are studied.Next,an estimate of the positive definite solutions is obtained.
In this paper,the Hermitian positive definite solution of the nonlinear matrix equation X~s+A~*X~(-t)A+B~*X~(-t)B=Q is investigated,in which A、Bare n×ncomplex matrices,s,t are positive integers and Qis a positive definite matrix.Firstly,based on linear system of equations theory,the property of the solutions are studied.Next,an estimate of the positive definite solutions is obtained.
引文
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[8]LIU X G,GAO H.On the positive definite solutions of the matrix equations Xs±A*X-t A=In[J].Linear Algebra Appl,2003(368):83-97.
[9]DUAN X,LIAO A.On the existence of Hermitian positive definite solutions of the matrix equation Xs+A*X-t A=Q[J].Linear Algebra Appl,2008(429):673-687.
[10]CAI J,CHEN G L.Some investigation on Hermitian positive definite solutions of matrix equation Xs+A*X-t A=Q[J].Linear Algebra Appl,2009(430):2448-2456.
[11]YANG Y T.The iterative method for solving nonlinear matrix equation Xs+A*X-t A=Q[J].Appl Math Comput,2007(188):46-53.
[12]ZHOU D M,CHEN G L,WU G X.Some properties of the nonlinear matrix equation Xs+A*X-t A=Q[J].J Math Anal Appl,2012(392):75-82.
[13]LONG J H,HU X Y,ZHANG L.On the Hermitian positive definite solutions of the nonlinear matrix equation X+A*X-1 A+B*X-1 B=I[J].Bull Braz Math Soc,New Series,2008(39):371-386.
[14]HE Y M,LONG J H.On the Hermitian positive definite solutions of nonlinear matrix equation X+∑m i=1A*iX-1 Ai=I[J].Comput Math Appl,2010(216):3480-3485.
[15]龙建辉.求解几类非线性矩阵方程的数值算法[D].长沙:湖南大学数学系,2008.
[16]ZHANG F Z.Matrix Theory:Basic Results and Techniques[M].Second.New York:Springer,2011.
[2]ENGWERDA J C.On the existence of a positive definite solution of the matrix equation X±A*X-1 A=I[J].Linear Algebra Appl,1993(194):91-108.
[3]ENGWERDA J C,RAN A C M,RIJKEBOER A L.Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equation X+A*X-1 A=Q[J].Linear Algebra and Its Application,1993(186):255-275.
[4]FERRANTE A,LEVY B.Hermitian solutions of the equation X=Q+NX-1 N*[J].Linear Algebra Appl,1996(247):359-373.
[5]BUZBEE B L,GOLUB G H,NIELSON C W.On direct methods for solving Poisson,s equations[J].SIAM J Numer Anal,1970(7):627-656.
[6]ZHAN X.Computing the extremal positive definite solutions of a matrix equation[J].SIAM J Sci Compute,1996(17):1167-1174.
[7]ZHAN X Z,XIE J J.On matrix equation[J].Linear Algebra Appl,1996(247):337-345.
[8]LIU X G,GAO H.On the positive definite solutions of the matrix equations Xs±A*X-t A=In[J].Linear Algebra Appl,2003(368):83-97.
[9]DUAN X,LIAO A.On the existence of Hermitian positive definite solutions of the matrix equation Xs+A*X-t A=Q[J].Linear Algebra Appl,2008(429):673-687.
[10]CAI J,CHEN G L.Some investigation on Hermitian positive definite solutions of matrix equation Xs+A*X-t A=Q[J].Linear Algebra Appl,2009(430):2448-2456.
[11]YANG Y T.The iterative method for solving nonlinear matrix equation Xs+A*X-t A=Q[J].Appl Math Comput,2007(188):46-53.
[12]ZHOU D M,CHEN G L,WU G X.Some properties of the nonlinear matrix equation Xs+A*X-t A=Q[J].J Math Anal Appl,2012(392):75-82.
[13]LONG J H,HU X Y,ZHANG L.On the Hermitian positive definite solutions of the nonlinear matrix equation X+A*X-1 A+B*X-1 B=I[J].Bull Braz Math Soc,New Series,2008(39):371-386.
[14]HE Y M,LONG J H.On the Hermitian positive definite solutions of nonlinear matrix equation X+∑m i=1A*iX-1 Ai=I[J].Comput Math Appl,2010(216):3480-3485.
[15]龙建辉.求解几类非线性矩阵方程的数值算法[D].长沙:湖南大学数学系,2008.
[16]ZHANG F Z.Matrix Theory:Basic Results and Techniques[M].Second.New York:Springer,2011.