Least-square solutions of inverse problems for reflexive matrices

Lianghua Liu
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Abstract

P = (pij) ∈ Cn×n is regarded as a generalized reflection matrix if P satisfies that PH = P, P2 = I. Let P ∈ Cn×n be a given generalized reflection matrix, A matrix A ∈ Cn×nis regarded as an n × n reflexive matrix with respect to P if A satisfies A = PAP. We denote the set of all n × n reflexive matrices by Crn×n(P). In this paper, the least-square solutions of the inverse problem of reflexive matrices is discussed, and the expression of the solution is obtained. In addition, the problem of using reflexive matrices to construct the optimal approximation to a given matrix is discussed, the necessary and sufficient conditions about the problem are derived, and the expression of the solutions is provided.
自反矩阵反问题的最小二乘解
若P满足PH = P, P2 = i,则将P = (pij)∈Cn×n视为广义反射矩阵,设P∈Cn×n为给定的广义反射矩阵,若a满足a = PAP,则将矩阵a∈Cn×nis视为关于P的n×n自反矩阵。我们用Crn×n(P)表示所有n×n自反矩阵的集合。本文讨论了自反矩阵反问题的最小二乘解,得到了其解的表达式。此外,讨论了用自反矩阵构造给定矩阵的最优逼近问题,导出了该问题的充要条件,并给出了其解的表达式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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