(R,S,μ)对称矩阵逆问题和最佳逼近问题及扰动分析

称R∈C^m×m为k次轮换矩阵若 R的最小多项式为x^k-1(k≥2).令μ∈{0,1,…,k-1}和ζ=e^2πi/k.若R∈C^m×m和S∈C^n×n为k次轮换矩阵,则称A∈C^m×m为(R,S,μ)对称矩阵若RAS^-1=ζ^μA.本文研究了(R,S,μ) 对称矩阵的逆问题和最佳逼近问题,得到了解的表达式. 并讨论了最佳逼近解的扰动分析,得到了比较满意的理论结果, 最后通过数值算例验证了该理论结果的正确性.

INVERSE PROBLEM AND PERTURBATION ANALYSIS ON THE APPROXIMATION PROBLEM FOR (R, S, μ) SYMMETRY MATRICES

A matrix R∈C^n×n is said to be k-involutary if its minimal polynomial is x^k-1 forsome k≥2.Letμ∈{0,1,...,k-1} andζ= e^2πi/k.If A∈C^m×n,R∈C^m×m,S∈C^n×nand R and S are k-involutory,we say that A is(R,S,μ-symmetric if RAS^-1 =ζ^μA.In this paper we discuss the inverse problem and associated approximation problem for(R,S,μ-symmetric matrices.Moreover,we provide a perturbation bound for the solution ofthe approximation problem and present some illustrative experiments to show the correctnessof the perturbation bound.