• 论文 •

### 矩阵方程X+A~*X~(-q)A=I(q>0)的Hermite正定解

1. 山东大学数学与系统科学学院,山东大学数学与系统科学学院,山东大学数学与系统科学学院 济南, 250100 ,济南, 250100 ,济南, 250100
• 出版日期:2004-01-14 发布日期:2004-01-14

### THE HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X + A~*X~(-q)A = I(q > 0)

1. Wang Jinfang Zhang Yuhai Zhu Benren (School of Mathematics and System Sciences, Shandong University, Jinan, 250100)
• Online:2004-01-14 Published:2004-01-14
1.引言 本文研究矩阵方程 X+A*X-qA=I (1)的Hermite正定解,其中I是一个n×n阶单位矩阵, A是一个n×n阶复矩阵, q是实数且q>0.q=1,q=2时的方程是从动态规划,随机过滤,控制理论和统计学中推导出来的,最近已有许多人对此进行了研究(见参考文献[1,2,4]),本文我们将研究方程(1)的解的存在性和解的性质,并讨论迭代求解及迭代解的收敛性. 对于Hermite矩阵X和Y,文中X≥Y表示X-Y是半正定的,X>y表示X-Y是正定的;对于方阵M,M*表示M的共轭转置,ρ(M)表示M的谱半径,λi(M)
We study the Hermitian positive definite solutions of the matrix equation X + A*X-qA = I with q > 0. Some properties of the solutions and the basic fixed point iterations for the equation are also discussed in some detail. Some of results in [Linear Algebra Appl., 279 (1998), 303-316], [Linear Algebra Appl, 326 (2001), 27-44] and [Linear Algebra Appl. 372 (2003), 295-304] are extended.
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 [1] I.G. Ivanov and S.M. El-Sayed, Properties of positive definite solution of the equation X + A*X-2A = I, Linear Algebra Appl, 279(1998) , 303-316． [2] I.G. Ivanov, V.I. Hasanov and B.V. Minchev, On matrix equation X ± A*X-2A-I, Linear Algebra Appl, 326(2001) , 27-44． [3] Yuhai Zhang, On Hermitian positive definite solutions of matrix equation X + A*X-2A = I, Linear Algebra Appl., 372(2003) , 295-304． [4] J.C. Engwerda, A.C.M. Ran, A.L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive deifnite solution of the matrix equation X + A*X-1A = Q, Linear Algebra Appl., 186(1993) , 255-275． [5] Rajendra Bhatia, Matrix Analysis, Springer-Verlag New York,INC., 1997． [6] G.W.Stewart and J.G.Sun, Matrix Perturbation Theory, Academic Press, 1990． [7] G.H.Golub and C.F.Van Loan, Matrix Computation, Johns Hopkins U.P., Baltimore, 1989． [8] Dajun Guo and V.Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, INC., 1988．
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