关于四元数体上某类矩阵方程的极小范数最小二乘解
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摘要
对四元数体上某类自共轭矩阵方程,在两两可交换的前提下,研究了矩阵方程的极小Frobenius范数最小二乘解.同时,在有解条件下给出了通解的表达形式.利用四元数体上自共轭矩阵奇异值分解的充分必要条件,运用四元数体上Frobenius范数正交矩阵乘积不变性,讨论了某类矩阵方程的最小二乘解,给出了极小Frobenius范数最小二乘解及其通解的表达形式,进而推广到了更为一般的矩阵方程.
For certain types self-conjugate matrix equation on quaternion field,we studied the minimal Frobenius norm least square solution of matrix equation under the premise of exchangeable,and gave the solution of general solution form under the condition of the solvability.By using sufficient and necessary conditions of the self-conjugate matrix singular value decomposition on quaternions,and invariance of Frobenius norm of orthogonal matrix product on quaternion,we discussed the least-square solutions of a kind of matrix equation,and give the expression of general solution form of the minimal Frobenius norm least squares solution,and extend to the more general matix equation in further.
For certain types self-conjugate matrix equation on quaternion field,we studied the minimal Frobenius norm least square solution of matrix equation under the premise of exchangeable,and gave the solution of general solution form under the condition of the solvability.By using sufficient and necessary conditions of the self-conjugate matrix singular value decomposition on quaternions,and invariance of Frobenius norm of orthogonal matrix product on quaternion,we discussed the least-square solutions of a kind of matrix equation,and give the expression of general solution form of the minimal Frobenius norm least squares solution,and extend to the more general matix equation in further.
引文
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[9]尤兴华,马圣容.关于矩阵方程组的最小二乘解和极小范数最小二乘解[J].苏州大学(自然科学版),2012,4(1):23-27.You Xinghua,Ma Shengrong.Least squares solution and minimum-norm least-squares solution of matrix equations[J].Journal of Soochow University(Natural Science Edition),2012,4(1):23-27.(in Chinese)
[10]庄瓦金.关于四元数矩阵的奇异值分解[J].新疆大学学报,1987,4(1):22-28.Zhuang Wajin.On the singular value decomposition for quaternion matrices[J].Journal of Xinjiang University(Natural Science Edition),1987,4(1):22-28.(in Chinese)
[2]Shifang Yuan,Anping Liao,Yuan Lei.Least squares Hermitian solution of the matrix equation with the least norm over the skew field of quaternions[J].Mathematical and Computer Modelling,2008,48:1-100.(in Chinese)
[3]王卿文,薛有才.体与环上的矩阵方程[M].北京:知识出版社,1996.
[4]王开明,刘永辉.四元数矩阵方程的最小二乘解[J].洛阳大学学报,2004,19(2):1-4.Wang Kaiming,Liu Yonghui.The Hermitian Leastsquare Solution of the Quaternion Matrix Equation[J].Journal of LuoYang university,2004,19(2):1-4.(in Chinese)
[5]刘永辉.四元数矩阵方程的最小二乘解[J].数学研究,2003,36(2):145-150.Liu Yonghui.The Least square Solutions to the Quaternion matrix Equation[J].Journal of Mathematical Study,2003,36(2):145-150.(in Chinese)
[6]韦刚和.关于四元数体上矩阵对角化的几个定理[J].常熟理工学院学报(自然科学版),2012,26(10):37-40.Wei Ganghe.Several Matrix Diagonalization Theorems on the Quaternion[J].Journal of Changshu Institute Technology(Natural Sciences).2012,26(10):37-40.(in Chinese)
[7]Loring T A.Factorization of matrices of quaternions[J].Expo.Math,2012,33(15):1-21.
[8]尤兴华,马圣容.李亚普诺夫方程的简洁解及其应用[J].南京师范大学报(自然科学版),2011,3(1):44-49.You Xinghua,Ma Shengrong.The Simple Formulae of Solutions to Liapunov Matrix Equation and Its Application[J].Journal of Nanjing Normal University(Natural Science Edition),2011,3(1):44-49.(in Chinese)
[9]尤兴华,马圣容.关于矩阵方程组的最小二乘解和极小范数最小二乘解[J].苏州大学(自然科学版),2012,4(1):23-27.You Xinghua,Ma Shengrong.Least squares solution and minimum-norm least-squares solution of matrix equations[J].Journal of Soochow University(Natural Science Edition),2012,4(1):23-27.(in Chinese)
[10]庄瓦金.关于四元数矩阵的奇异值分解[J].新疆大学学报,1987,4(1):22-28.Zhuang Wajin.On the singular value decomposition for quaternion matrices[J].Journal of Xinjiang University(Natural Science Edition),1987,4(1):22-28.(in Chinese)