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线性流形上的广义中心对称矩阵反问题

袁永新,戴华   

  1. 南京航空航天大学理学院,南京航空航天大学理学院 南京 210016 华东船舶工业学院数理系,镇江 212003 ,南京 210016
  • 出版日期:2005-04-14 发布日期:2005-04-14

袁永新,戴华. 线性流形上的广义中心对称矩阵反问题[J]. 计算数学, 2005, 27(4): 383-394.

INVERSE PROBLEMS OF GENERALIZED CENTROSYMMETRIC MATRICES ON THE LINEAR MANIFOLD

  1. Yuan Yongxin (College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; Dept. of Mathematics and Physics, East China Shipbuilding Institute, Zhenjiang 212003, China) Dai Hua (College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)
  • Online:2005-04-14 Published:2005-04-14

设R∈Cn×n是满足R=RH=R-1≠±In的广义反射矩阵.若A∈Cn×n满足RAR=A,则称A为n阶广义中心对称矩阵,n阶广义中心对称矩阵的全体记为GCSCn×n.令X1,Z1∈Cn×k1,Y1,W1∈Cn×l1,S={A|‖AX1-Z1‖2+‖Y1HA-W1H‖2=min,A∈GCSCn×n},本文研究如下问题.问题Ⅰ.给定矩阵Z2,X2∈Cn×k2,Y2,W2∈Cn×l2,求A∈S,使得其中‖·‖是Frobenius范数.问题Ⅱ.给定矩阵A∈Cn×n,求A∈SE,使得其中SE是问题Ⅰ的解集合.本文给出了问题Ⅰ解集合SE的表达式,并导出了矩阵方程AX2=Z2,Y2HA=W2H有解A∈S的充分必要条件及其通解表达式,并给出了问题Ⅱ解的表达式以及求解问题Ⅱ的数值方法和数值例子.

Let R∈Cn×n satisfying R = RH=R-1≠±In be a nontrivial generalized reflexive matrix. A∈Cn×n is said to be generalized centrosymmetric if RAR = A. The set of all n×n generalized centrosymmetric matrices is denoted by GCSCn×n. Let X1,Z1∈Cn×k1,Y1,W1∈Cn×l1,S = {A|‖AX1-Z1‖2+‖Y1HA-W1H‖2= min, A∈GCSCn×n}. The following problems are considered. Problem Ⅰ. Given Z2,X2∈ Cn×k2;Y2,W2 ∈Cn×l2, find A∈S such that where ‖·‖ is the Frobenius norm. Problem Ⅱ. Given A∈Cn×n, find A ∈ SE such that where SE is the solution set of Problem Ⅰ. The general form of the solution set SE of Problem Ⅰ is given. Sufficient and necessary conditions for matrix equations AX2=Z2,Y2HA = W2H having a solution A∈S are derived, and the general solutions are given. The expression of the solution to Problem Ⅱ is presented. A numerical example is provided.

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