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INVERSE PROBLEMS OF GENERALIZED CENTROSYMMETRIC MATRICES ON THE LINEAR MANIFOLD

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)

Abstract:

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.

Keywords: generalized centrosymmetric matrix, inverse problem, linear manifold, optimal approximation

DOI: