线性约束下Hermite-广义反Hamilton矩阵的最佳逼近问题

本文利用对称向量与反对称向量的特征性质,给出了约束矩阵集合非空的充分必要条件及矩阵的一般表达式.运用空间分解理论和闭凸集上的逼近理论,得到了任一n阶复矩阵在约束矩阵集合中的惟一最佳逼近解.

THE OPTIMAL APPROXIMATION PROBLEMS OF HERMITE-GENERALIZED ANTIHAMILTION MATRICES UNDER THE LINEAR RESTRICTION

Let $OASR^{ntimes n}={J|J^{rm T}J=JJ^{rm T}=I_n$, $J=-J^{rm T}$,$Jin R^{ntimes n}}$. Given $Jin OASR^{ntimes n}, Ain C^{ntimes n}$ is termed generalized Hamiltonmatrix if $JAJ=A^H$. We denote the set of all $ntimes n$ generalized Hamiltonmatrices by $HTC^{ntimes n}$. $Ain C^{ntimes n}$ is termed generalized Antihamilton matrix if $JAJ=-A^H$.We denote the set of all $ntimes n$ generalized Antihamilton matrices by$AHTC^{ntimes n}$ $Ain C^{ntimes n}$ is termed Hermite-genderalized Antihamilton matrix if$$ A^H=H mbox{and} JAJ=-A^H.$$ We denote the set of all $ntimes n$ Hermite-generalized antihamiltonmatrices by $HAHC^{ntimes n}$. In this paper, we discuss the following two problems. {bf Problem bf I}~~ Given $X, Bin C^{ntimes m}$, Find $Ain HAHC^{ntimes n}$ such that $AX=B$. {bf Problem bf II}~~ Given $A^*in C^{ntimes n}$, Find $widehat{A}in S_A$ such that$$ |A^*-widehat{A}|=minlimits_{Ain S_A}|A^*-A|,$$ where $|cdot|$ is Frobenius norm, and $S_A$ is the solution of Problem I. In this paper, the general representation of $S_A$ has been given. The necessaryand sufficient condition have been presented for Problem I. For Problem II the expression of the solution has been provided.