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 $OASR^{n\times n}=\{J|J^{\rm T}J=JJ^{\rm T}=I_n$, $J=-J^{\rm T}$, $J\in R^{n\times n}\}$. Given $J\in OASR^{n\times n}, A\in C^{n\times n}$ is termed generalized Hamilton matrix if $JAJ=A^H$. We denote the set of all $n\times n$ generalized Hamilton matrices by $HTC^{n\times n}$. $A\in C^{n\times n}$ is termed generalized Antihamilton matrix if $JAJ=-A^H$. We denote the set of all $n\times n$ generalized Antihamilton matrices by $AHTC^{n\times n}$ $A\in C^{n\times n}$ is termed Hermite-genderalized Antihamilton matrix if $$ A^H=H \ \ \ \ \ \mbox{and} \ \ \ \ \ JAJ=-A^H. $$ We denote the set of all $n\times n$ Hermite-generalized antihamilton matrices by $HAHC^{n\times n}$. In this paper, we discuss the following two problems. {\bf Problem \bf I}~~ Given $X, B\in C^{n\times m}$, Find $A\in HAHC ^{n\times n}$ such that $AX=B$. {\bf Problem \bf II}~~ Given $A^*\in C^{n\times n}$, Find $\widehat{A} \in S_A$ such that $$ \|A^*-\widehat{A}\|=\min\limits_{A\in S_A}\|A^*-A\|, $$ 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 necessary and sufficient condition have been presented for Problem I. For Problem II the expression of the solution has been provided.