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反对称正交反对称矩阵反问题的最小二乘解
引用本文:周富照,赵人可. 反对称正交反对称矩阵反问题的最小二乘解[J]. 黑龙江大学自然科学学报, 2004, 21(4): 79-84
作者姓名:周富照  赵人可
作者单位:武汉大学,数学与统计学院,湖北,武汉,430072;长沙理工大学,数学与计算科学学院,湖南,长沙,410076;长沙理工大学,数学与计算科学学院,湖南,长沙,410076
基金项目:Supported by the Natural Science Foundation of China under (10171031,50208004)
摘    要:设P为一给定的对称正交矩阵,记AAnp={A∈Rn×n‖AT=-A,(PA)T=-PA}.讨论下列问题问题Ⅰ给定X,B∈Rn×m.求A∈AARnp使‖AX-B‖=min.问题Ⅱ设A∈Rn×n,求A*∈SE使‖A-A*‖=infA∈SE‖A-A‖,其中SE为问题Ⅰ的解集合,‖·‖表示Frobenius范数.研究AARnp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*,且得到了此解的具体表达式.

关 键 词:反对称正交反对称矩阵  最小二乘解  最佳逼近

Least-square solutions of inverse problems for anti-symmetric ortho-antisymmetric matrices
ZHOU Fu-zhao,ZHAO Ren-ke. Least-square solutions of inverse problems for anti-symmetric ortho-antisymmetric matrices[J]. Journal of Natural Science of Heilongjiang University, 2004, 21(4): 79-84
Authors:ZHOU Fu-zhao  ZHAO Ren-ke
Abstract: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 following two problems are discussed in this paper:Problem Ⅰ: Given X, B ∈ Rn×m, find A ∈ AARnp such that f(A) = ‖AX - B‖ = min,Problem Ⅱ: Given A ∈ Rn×n, find A* ∈ SE such that ‖A~-A*‖=A∈SE‖A~-A‖ where ‖·‖ is the Frobenius norm, and SE is given. For Problem Ⅱ, the expression of the solution is provided. Futhermore, it is pointed that some results of References 2 and 7 are special cases of this paper.
Keywords:anti-symmetric ortho-antisymmetric matrix  least-square solution  optimal approximation  
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