线性流形上Hermite-广义反Hamilton矩阵反问题的最小二乘解

LEAST-SQUARES SOLUTIONS OF INVERSE PROBLEMS FOR HERMITE-GENERALIZED ANTIHAMILTON MATRICES ON THE LINEAR MANIFOLD

1.引言 令Rn×m表示所有n×m实矩阵集合,Cn×m表示所有n×m复矩阵集合,Cn=Cn×1,HCn×n表示所有n阶Hermite矩阵集合,UCn×n表示所有n阶酉矩阵集合,AHCn×n表示所有n阶反Hermite矩阵集合,R(A)表示A的列空间,N(A)表示A的零空间,A+表示A的Moore—Penrose广义逆,A*B表示A与B的Hadamard积,rank(A)表示矩阵A的秩.tr(A)表示矩阵A的迹.矩阵A,B的内积定义为(A,B)=tr(BHA),A,B∈Cn×m,由此内积诱导的范数为||A||=√(A,A)=[tr(AHA)]1/2,则此范数为Frobenius范数,并且Cn×m构成一个完备的内积空间,In表示n阶单位阵,i=√-1,记OASRn×n表示n×n阶正交反对称矩阵的全体。

Let OASR = Given J OASR,A C is termed generalized Hamilton matrix if JAT = AH. we denote the set of all n x n generalized Hamilton matrics by HTC, AC is termed generalized antihamilton matrix if JAJ = -AH. We denote the set of all n x n generalized antihamilton matrices by AHTC. A 6 C is termed Hermite-generalized antihamilton matrix if AH = A and JAJ = -AH. We denote the set of all n x n Hermite-generalized antihamilton matrices by Let Where U = (U1, U2) UC, In this paper, we discuss the following two problems: Problem I. Given X,B C, Find A S such that Problem II. Given A* C, Find A SE such that Where is Frobenius norm, and SE is the solution set of Problem I. The general representation of SE has been given. The necessary and sufficient conditions have been presented for f(A) = 0. For Problem II the expression of the solution has been provided.