反对称正交反对称矩阵反问题的最小二乘解(英文)

Least-square solutions of inverse problems for anti-symmetric ortho-antisymmetric matrices

设P为一给定的对称正交矩阵, 记AARP = {A ∈ Rn n ×n AT = ?A,(PA)T = ?PA}. 讨论下列问题:问题Ⅰ给定X,B ∈ Rn ×m . 求A ∈ AARP使 AX ? B = min. n问题Ⅱ设A? ∈ Rn , 求A? ∈ SE 使 A? ? A? = infA ×n ∈SE A? ? A , 其中SE为问题Ⅰ的解集合, · 表示Frobenius范数. 研究AARP中元素的通式, 给出问题Ⅰ解的一般表达式, 证明了问题Ⅱ存在唯一逼近解A?, 且 n得到了此解的具体表达式.

Given P ∈ ORn ×n satisfying PT = P. A ∈ Rn ×n is called anti-symmetric ortho-antisymmetric matrix if A = ?AT, (PA)T = ?PA. The set of all n × n anti-symmetric ortho-antisymmetric matrices is denoted by AARP. The n following two problems are discussed in this paper: Problem I: Given X,B ∈ Rn , ?nd A ∈ AARP such that f(A) = AX ? B = min, ×m n Problem II: Given A ∈ Rn , ?nd A? ∈ SE such that e ×n A ? A? = inf e A ? A e A∈SE where · is the Frobenius norm,and SE is the solution set of Problem I. For Problem I, the general form of SE is given. For Problem II, the expression of the solution is provided. Further- more, it is pointed that some results of References 2 and 7 are special cases of this paper.