反对称自正交相似矩阵反问题的解

The Solution for Inverse Problem of Anti-Symmetric and Self-Orthogonal Similar Matrices

设R为实数域,A∈R2k×2k, J=[0Sk-Sk0],若JAJT=A,AT=-A,则称A为反对称自正交相似矩阵.全体n阶反对称自正交相似矩阵的集合记为A Jn×n,n=2k.本文研究了如下反对称自正交相似矩阵反问题:问题Ⅰ: 己知X、B∈Rn×m,求A∈AJn×n,使得AX=B;问题Ⅱ: 已知A*∈Rn×n,求(A)∈SE,使得‖A*-(A)‖=infA∈SE‖A*-A‖.其中SE是问题Ⅰ的解集.给出了问题Ⅰ解存在的条件及一般解的表达式,也给出了问题Ⅱ的唯一解.

Let J=[0 Sk -Sk 0],A∈R2k×2k,if JAJT=AT,AT=-A,then A is called anti-symmetric and self-orthogonal similar matrix.The set of anti-symmetric and self-orthogonal similar matrix is denoted AJn×n.This paper discusses the following problems:ProblemⅠ: Give X、B∈Rn×m,find A∈AJn×nsuch that AX=B.ProblemⅡ: Give A*∈Rn×n,find A～∈SE such that‖A*-‖=infA∈SE‖A*-A‖,where SE is the solution set of ProblemⅠ.The existence of the solution for ProblemⅠand uniquness of the solution for Problem Ⅱare proved.The general form of SE is given and the expression of  is presented.