基于正交投影方法的二次特征值反问题及其最佳逼近解
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摘要
考虑二次特征值反问题的广义中心对称解(广义反中心对称解)及其最佳逼近问题,应用矩阵的正交投影方法,给出矩阵方程AX+BY+CZ=0的解及其最佳逼近问题.利用广义中心对称矩阵(广义反中心对称矩阵)的性质导出了该问题有广义中心对称解(广义反中心对称解)的条件及有解情况下的通解表达式,并证明了最佳逼近问题解的存在性与唯一性,得到了最佳逼近解的表达式.
We considered the generalized centrosymmetric solution(generalized anti-centrosymmetric solution)of an inverse quadratic eigenvalue problem and its optimal approximation problem.By using the orthogonal projection methods of matrix,we gave the solution of matrix equation AX+BY+CZ=0and its optimal approximation problem.According to the properties of generalized centrosymmetric matrices(generalized anti-centrosymmetric matrices),we derived the conditions for the problem with ageneralized centrosymmetric solution(generalized anti-centrosymmetric solution)and the expression of general solution. We proved the existence and the uniqueness of solution of the optimal approximation problem,and obtained the expression of the optimal approximation solution.
We considered the generalized centrosymmetric solution(generalized anti-centrosymmetric solution)of an inverse quadratic eigenvalue problem and its optimal approximation problem.By using the orthogonal projection methods of matrix,we gave the solution of matrix equation AX+BY+CZ=0and its optimal approximation problem.According to the properties of generalized centrosymmetric matrices(generalized anti-centrosymmetric matrices),we derived the conditions for the problem with ageneralized centrosymmetric solution(generalized anti-centrosymmetric solution)and the expression of general solution. We proved the existence and the uniqueness of solution of the optimal approximation problem,and obtained the expression of the optimal approximation solution.
引文
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[12]张贤达.矩阵分析与应用[M].2版.北京:清华大学出版社,2013.(ZHANG Xianda.Matrix Analysis and Application[M].2nd ed.Beijing:Tsinghua University Press,2013.)
[2]Chu M T,Kuo Y C,LIN Wenwei.On Inverse Quadratic Eigenvalue Problems with Partially Prescribed Eigenstructive[J].SIAM Journal on Matrix Analysis and Applications,2004,25(4):995-1020.
[3]桂冰,戴华.一类二次特征值反问题的中心对称解及其最佳逼近[J].高等学校计算数学学报,2006,28(4):367-373.(GUI Bing,DAI Hua.The Centrosymmetric Solution of the Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation[J].Numer Math J Chinese Univ,2006,28(4):367-373.)
[4]梁俊平,卢琳璋.二次特征值反问题的中心斜对称解及其最佳逼近[J].福建师范大学学报(自然科学版),2006,22(3):10-14.(LIANG Junping,LU Linzhang.The Centroskew Symmetric Solution of the Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation[J].Journal of Fujian Normal University(Natural Science Edition),2006,22(3):10-14.)
[5]CAI Yunfeng,XU Shufang.On a Quadratic Inverse Eigenvalue Problem[J].Inverse Problems,2009,25(8):085004.
[6]YUAN Yongxin,DAI Hua.On a Class of Inverse Quadratic Eigenvalue Problem[J].Journal of Computational and Applied Mathematics,2011,235(8):2662-2669.
[7]YUAN Yongxin,DAI Hua.Solutions to an Inverse Monic Quadratic Eigenvalue Problem[J].Linear Algebra and Its Applications,2011,434(11):2367-2381.
[8]郭丽杰,周硕.二次特征值反问题的对称次反对称解及其最佳逼近[J].吉林大学学报(理学版),2009,47(6):1185-1190.(GUO Lijie,ZHOU Shuo.Symmetric and Skew Anti-symmetric Solution of Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation[J].Journal of Jilin University(Science Edition),2009,47(6):1185-1190.)
[9]周硕,韩明花,孟欢欢.用试验数据修正振动系统的双对称阻尼矩阵与刚度矩阵[J].应用数学和力学,2014,35(6):697-711.(ZHOU Shuo,HAN Minghua,MENG Huanhuan.Bisymmetric Damping and Stiffness Matrices Calibration with Test Data of Vibration Systems[J].Applied Mathematics and Mechanics,2014,35(6):697-711.)
[10]吕晓寰,程宏伟,方彬彬,等.基于商奇异值分解的一类二次特征值反问题[J].东北电力大学学报,2015,35(1):88-92.(LXiaohuan,CHENG Hongwei,FANG Binbin,et al.On a Class of Inverse Quadratic Eigenvalue Problem Based on Quotient Singular Value Decomposition[J].Journal of Northeast Dianli University,2015,35(1):88-92.)
[11]陈景良,陈向晖.特殊矩阵[M].北京:清华大学出版社,2001.(CHEN Jingliang,CHEN Xianghui.Special Matrix[M].Beijing:Tsinghua University Press,2001.)
[12]张贤达.矩阵分析与应用[M].2版.北京:清华大学出版社,2013.(ZHANG Xianda.Matrix Analysis and Application[M].2nd ed.Beijing:Tsinghua University Press,2013.)