几类约束矩阵方程问题的研究
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摘要
约束矩阵方程问题广泛应用于自动控制、振动理论、系统参数识别以及非线性规划等领域,本文分别从递推算法以及利用奇异值分解、标准相关分解和广义奇异值分解的直接算法两个不同角度系统地研究几类约束矩阵方程的求解问题:
问题Ⅰ 给定X,B∈R~(n×m)及某类矩阵集合S(?)R~(n×m),求A∈S使得AX=B.
问题Ⅱ 给定X,B∈R~(n×m)及某类矩阵集合S(?)R~(n×m),求A∈S使得‖AX-B‖=min
问题Ⅲ 给定A∈R~(n×m),B∈R~(n×m)及某类矩阵集合S(?)R~(n×m),求X∈S使得‖A~TXA-B‖=min
问题Ⅳ 给定(?)∈R~(n×m),求(?)∈S_E使得(?),其中S_E表示问题Ⅰ、Ⅱ或Ⅲ的解集合,‖·‖表示矩阵的Frobenius范数。
本文的主要结果如下:
1.研究了矩阵方程AX+XB=F对称解的递推算法,该算法不仅能够用于对称解存在性的判断问题,而且当对称解存在时,也能够用于对称解的计算问题,选取特殊的初始矩阵时,该算法还能够得到矩阵方程的极小范数对称解;随后讨论了矩阵方程AXB=C反对称解的递推算法。
2.当S是对称正交反对称矩阵集合时,给出了问题Ⅰ有解的充要条件、解的一般表达式以及相应问题Ⅳ的解;对于S={A∈SAR_P~n|‖AZ-Y‖=min},给出了问题Ⅱ的解以及相应问题Ⅳ解的表达式;利用矩阵对的标准相关分解还给出了问题Ⅲ的解。
3.对于s={A∈AAR_P~n|AZ=Y,Y_i~TZ_i=-Z_i~TY_i,Y_iZ_i~+Z_i=Y_i,i=1,2},给出了问题Ⅱ解的一般表达式,相应问题Ⅳ也得到了解决;当S是反对称正交反对称矩阵集合时,利用矩阵对的标准相关分解给出了问题Ⅲ解的一般表达式。
4.当S是反对称正交对称矩阵集合时,给出了问题Ⅰ有解的充要条件、解的一般表达式以及相应问题Ⅳ的解;利用矩阵对的广义奇异值分解,给出了矩阵方程A~TXA=B反对称正交对称解存在的充要条件、解的一般表达式以及相应问题Ⅳ的解;对于S={A∈ASR_P~n|‖AZ—Y‖=min},给出了问题Ⅱ的解以及相应问题Ⅳ解的表达式。
The constrained matrix equation problems have been widely used in many fields such as control theory, vibration theory, system parameters identification, nonlinear programming and so on. In this paper, we will systematically study several kinds of constrained matrix equation problems from different aspects: recursive algorithm and direct calculation method by using singular value decomposition, the canonical singular value decomposition and the generalized singular value decomposition of matrices. The main problems discussed are as follows:Problem Ⅰ Given X,B∈ R~(n×m) and the matrices set S (?) R~(n×n), find A∈S such that AX = B.Problem Ⅱ Given X, B ∈ R~(n×m) and the matrices set S (?) R~(n×n), find A ∈ Ssuch that ||AX - B|| = min .Problem Ⅲ Given A ∈ R~(n×m), B ∈ R~(m×m) and the matrices set S (?) R~(n×n), find X∈S such that||A~TXA-B|| = min.Problem Ⅳ Given (?) ∈ R~(n×n) , find A~* ∈ S_E such thatwhere S_E is the solution set of Problem Ⅰ , Ⅱ or Ⅲ and ||·|| is the Frobeniusnorm.The main results of this paper are as follows:1. A recursive algorithm to solve matrix equation AX + XB = F over symmetric solutions is constructed. By this algorithm, the solvability of the equation over symmetric solutions can be determined. When the matrix equation is consistent, the symmetric solutions can be obtained and its least-norm symmetric solution can be given by choosing a special initial matrix. The recursive algorithm to solve matrix equation AXB = C over anti-symmetric solutions is also discussed.2. We present the sufficient and necessary conditions for Problem I and give the expressions of solutions for Problem Ⅰ and Problem Ⅳ when S is the set of all symmetric orth-symmetric matrices. The expressions of the solutions for Problem Ⅱ and the corresponding problem Ⅳ are get, in which S can be written asS = {A ∈ SAR_p~n| ||AZ - Y|| = min}. In addition, by using the canonical singular value decomposition, the expressions of solutions for Problem Ⅲ is solved.3. Over the linear manifold S = {A ∈ AAR_p~n| AZ = Y,Y_i~TZ_i = -Z_i~TY_i,Y_iZ_i~+Z_i =
Yt,i = 1,2}, the expressions of solutions for Problem II and the correspondingproblem IV are given. When S is the set of all anti-symmetric orth-aniti-symmetric matrices, the expressions of solutions for Problem III is derived by the canonical singular value decomposition of matrices.4. The sufficient and necessary conditions for Problem I and the expressions of solutions for Problem I and Problem IV are given when S is the set of all anti-symmetric orth-symmetric matrices. By applying the generalized singular value decomposition of matrices, we derive the necessary and sufficient conditions for the existence of the anti-symmetric orth-symmetric solution of linear matrix equation AJXA = B . The general expression of the solutions for A^XA-B and its corresponding problem IV are given. In addition, the expressions of the solutions for Problem II and the corresponding problem IV are provided over the set S whichcan be written as S = \A e ASR'^, AZ - F|j =
问题Ⅰ 给定X,B∈R~(n×m)及某类矩阵集合S(?)R~(n×m),求A∈S使得AX=B.
问题Ⅱ 给定X,B∈R~(n×m)及某类矩阵集合S(?)R~(n×m),求A∈S使得‖AX-B‖=min
问题Ⅲ 给定A∈R~(n×m),B∈R~(n×m)及某类矩阵集合S(?)R~(n×m),求X∈S使得‖A~TXA-B‖=min
问题Ⅳ 给定(?)∈R~(n×m),求(?)∈S_E使得(?),其中S_E表示问题Ⅰ、Ⅱ或Ⅲ的解集合,‖·‖表示矩阵的Frobenius范数。
本文的主要结果如下:
1.研究了矩阵方程AX+XB=F对称解的递推算法,该算法不仅能够用于对称解存在性的判断问题,而且当对称解存在时,也能够用于对称解的计算问题,选取特殊的初始矩阵时,该算法还能够得到矩阵方程的极小范数对称解;随后讨论了矩阵方程AXB=C反对称解的递推算法。
2.当S是对称正交反对称矩阵集合时,给出了问题Ⅰ有解的充要条件、解的一般表达式以及相应问题Ⅳ的解;对于S={A∈SAR_P~n|‖AZ-Y‖=min},给出了问题Ⅱ的解以及相应问题Ⅳ解的表达式;利用矩阵对的标准相关分解还给出了问题Ⅲ的解。
3.对于s={A∈AAR_P~n|AZ=Y,Y_i~TZ_i=-Z_i~TY_i,Y_iZ_i~+Z_i=Y_i,i=1,2},给出了问题Ⅱ解的一般表达式,相应问题Ⅳ也得到了解决;当S是反对称正交反对称矩阵集合时,利用矩阵对的标准相关分解给出了问题Ⅲ解的一般表达式。
4.当S是反对称正交对称矩阵集合时,给出了问题Ⅰ有解的充要条件、解的一般表达式以及相应问题Ⅳ的解;利用矩阵对的广义奇异值分解,给出了矩阵方程A~TXA=B反对称正交对称解存在的充要条件、解的一般表达式以及相应问题Ⅳ的解;对于S={A∈ASR_P~n|‖AZ—Y‖=min},给出了问题Ⅱ的解以及相应问题Ⅳ解的表达式。
The constrained matrix equation problems have been widely used in many fields such as control theory, vibration theory, system parameters identification, nonlinear programming and so on. In this paper, we will systematically study several kinds of constrained matrix equation problems from different aspects: recursive algorithm and direct calculation method by using singular value decomposition, the canonical singular value decomposition and the generalized singular value decomposition of matrices. The main problems discussed are as follows:Problem Ⅰ Given X,B∈ R~(n×m) and the matrices set S (?) R~(n×n), find A∈S such that AX = B.Problem Ⅱ Given X, B ∈ R~(n×m) and the matrices set S (?) R~(n×n), find A ∈ Ssuch that ||AX - B|| = min .Problem Ⅲ Given A ∈ R~(n×m), B ∈ R~(m×m) and the matrices set S (?) R~(n×n), find X∈S such that||A~TXA-B|| = min.Problem Ⅳ Given (?) ∈ R~(n×n) , find A~* ∈ S_E such thatwhere S_E is the solution set of Problem Ⅰ , Ⅱ or Ⅲ and ||·|| is the Frobeniusnorm.The main results of this paper are as follows:1. A recursive algorithm to solve matrix equation AX + XB = F over symmetric solutions is constructed. By this algorithm, the solvability of the equation over symmetric solutions can be determined. When the matrix equation is consistent, the symmetric solutions can be obtained and its least-norm symmetric solution can be given by choosing a special initial matrix. The recursive algorithm to solve matrix equation AXB = C over anti-symmetric solutions is also discussed.2. We present the sufficient and necessary conditions for Problem I and give the expressions of solutions for Problem Ⅰ and Problem Ⅳ when S is the set of all symmetric orth-symmetric matrices. The expressions of the solutions for Problem Ⅱ and the corresponding problem Ⅳ are get, in which S can be written asS = {A ∈ SAR_p~n| ||AZ - Y|| = min}. In addition, by using the canonical singular value decomposition, the expressions of solutions for Problem Ⅲ is solved.3. Over the linear manifold S = {A ∈ AAR_p~n| AZ = Y,Y_i~TZ_i = -Z_i~TY_i,Y_iZ_i~+Z_i =
Yt,i = 1,2}, the expressions of solutions for Problem II and the correspondingproblem IV are given. When S is the set of all anti-symmetric orth-aniti-symmetric matrices, the expressions of solutions for Problem III is derived by the canonical singular value decomposition of matrices.4. The sufficient and necessary conditions for Problem I and the expressions of solutions for Problem I and Problem IV are given when S is the set of all anti-symmetric orth-symmetric matrices. By applying the generalized singular value decomposition of matrices, we derive the necessary and sufficient conditions for the existence of the anti-symmetric orth-symmetric solution of linear matrix equation AJXA = B . The general expression of the solutions for A^XA-B and its corresponding problem IV are given. In addition, the expressions of the solutions for Problem II and the corresponding problem IV are provided over the set S whichcan be written as S = \A e ASR'^, AZ - F|j =
引文
[1] Peng Yaxin, Hu Xiyan, Zhanglei. An iteration methods for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C. Applied Mathematics and Computation. 2005, 160: 763-777
[2] Deng Yuanbei, Hu Xiyan, Zhanglei. The solvability conditions for the inverse eigenvalue problem of the symmetrizable matrices. Journal of Computational and Applied Mathematics. 2004, 163: 101-106
[3] Zhou Fuzhao, Hu Xiyan, Zhanglei. The solvability conditions for the inverse problems of symmetric ortho-symmetric matrices. Applied Mathematics and Computation. 2004, 154: 153-166
[4] Peng Zhenyun, Hu Xiyan. The reflexive and anti-reflexive solutions of the matrix equation AX=B. Linear Algebra and its Applications. 2003, 375: 147-155
[5] Allwright J. C. Positive semidefinite matrices: Characterization Via conical hulls and least-squares solution of a matrix equation. SIAM J. Control Optima. 1998, 26: 537-556
[6] Xu Guiping, Wei Musheng and Zheng Daosheng. On solutions of matrix equation AXB+CYD=F. Linear Algebra and its Applications. 1998, 279: 93-109
[7] Nian. Li. A matrix inverse eigenvalue problems and its application. Linear Algebra and its Applications. 1997, 266: 143-152
[8] Dai hua, P.Lancaster. Linear matrix equation from an inverse problem of vibration theory. Linear Algebra and its Applications. 1996, 246: 31-47
[9] Nian. Li and K.-W. E. Chu. Designing the Hopfield neural network Via pole assignment. International Journal Systems Science. 1994, 25: 669-681
[10] Gloub, G.H.and Zha, H.Y.. Perturbation analysis of the canonical correlations of matrix pairs. Linear Algebra and its Applications. 1994, 210: 3-28
[11] Jameson A., Kreindler E. and Lancaster R. Symmetric Positive semidefinite and positive definite real solution of AX=XA~T and AX=YB. Linear Algebra and its Applications. 1992, 160: 189-215
[12] Allwright J. C. and Woodgate K. G Errata and Addendum to: Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation. SLAM. J. Control Optima. 1990, 28: 250-251
[13] Wei F. S. Analytical dynamic model improvement using vibration test data. AIAA J. 1990, 28: 175-177
[14] Henk Don F J. On the symmetric solutions of a linear matrix equation. Linear Algebra and its Applications. 1987, 93: 1-7
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[18] Baruch M. Optimal correction of mass and stiffness matrices using measured modes. AIAA J. 1982, 20(11): 1623-1626
[19] J.K. John Jones and C. Lew. Solutions of Liapuonv matrix equation BX-XA=C. IEEE Trans. Automatic Control. 1982, AC-27: 464-466
[20] Stewart, G. W.. Computing the CS-decomposition of a partitioned orthogonal matrix. Number.Math.. 1982, 40: 297-306
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[22] Paige, C. C. and Saunders, M. A.. Towards a generalized singular value decomposition. SIAM J. Numer. Anal.. 1981, 18: 398-405
[23] Wei F. S. Stiffness matrix correction from incomplete test data. AIAA J. 1980, 18(10): 1274-1275
[24] Berman A. Mass matrix correction using an incomplete set of measure modes. AIAA J. 1979, 17: 1147-1148
[25] Baruch M. Optimization procedure to correct stiffness and flexibility matrices using vibration tests. AIAA J. 1978, 16(11): 1208-1210
[26] Baruch M, Bar Itzhack IY. Optimal weighted orthogonalization of measured modes. AIAA J. 1978, 16(4): 345-351
[27] Jameson A. and Kreindler E. Inverse problem of linear Optimal control. SIAM J. Control. 1973, 11: 1-19
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[29] Brock I. E. Optimal matrices describing linear system. AIAA J. 1968, 6: 1292-1296
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[35] 彭亚新,胡锡炎,张磊.矩阵方程A~TXA=B的对称正交对称解及其最佳逼近.高等学校计算数学学报.2003,25(4):372-377
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[2] Deng Yuanbei, Hu Xiyan, Zhanglei. The solvability conditions for the inverse eigenvalue problem of the symmetrizable matrices. Journal of Computational and Applied Mathematics. 2004, 163: 101-106
[3] Zhou Fuzhao, Hu Xiyan, Zhanglei. The solvability conditions for the inverse problems of symmetric ortho-symmetric matrices. Applied Mathematics and Computation. 2004, 154: 153-166
[4] Peng Zhenyun, Hu Xiyan. The reflexive and anti-reflexive solutions of the matrix equation AX=B. Linear Algebra and its Applications. 2003, 375: 147-155
[5] Allwright J. C. Positive semidefinite matrices: Characterization Via conical hulls and least-squares solution of a matrix equation. SIAM J. Control Optima. 1998, 26: 537-556
[6] Xu Guiping, Wei Musheng and Zheng Daosheng. On solutions of matrix equation AXB+CYD=F. Linear Algebra and its Applications. 1998, 279: 93-109
[7] Nian. Li. A matrix inverse eigenvalue problems and its application. Linear Algebra and its Applications. 1997, 266: 143-152
[8] Dai hua, P.Lancaster. Linear matrix equation from an inverse problem of vibration theory. Linear Algebra and its Applications. 1996, 246: 31-47
[9] Nian. Li and K.-W. E. Chu. Designing the Hopfield neural network Via pole assignment. International Journal Systems Science. 1994, 25: 669-681
[10] Gloub, G.H.and Zha, H.Y.. Perturbation analysis of the canonical correlations of matrix pairs. Linear Algebra and its Applications. 1994, 210: 3-28
[11] Jameson A., Kreindler E. and Lancaster R. Symmetric Positive semidefinite and positive definite real solution of AX=XA~T and AX=YB. Linear Algebra and its Applications. 1992, 160: 189-215
[12] Allwright J. C. and Woodgate K. G Errata and Addendum to: Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation. SLAM. J. Control Optima. 1990, 28: 250-251
[13] Wei F. S. Analytical dynamic model improvement using vibration test data. AIAA J. 1990, 28: 175-177
[14] Henk Don F J. On the symmetric solutions of a linear matrix equation. Linear Algebra and its Applications. 1987, 93: 1-7
[15] Mitra S.K. The matrix equations AX=C, XB=D. Linear Algebra and its Applications. 1984, 59: 171-181
[16] Berman A, Nagy E. J. Improvement of a large analytical model using test data. AIAA J. 1983, 21(8): 1168-1173
[17] Golub, C.H. and Van Loan, C.F.. Matrix computations. The Johns Hopkins University Press, American, 1983
[18] Baruch M. Optimal correction of mass and stiffness matrices using measured modes. AIAA J. 1982, 20(11): 1623-1626
[19] J.K. John Jones and C. Lew. Solutions of Liapuonv matrix equation BX-XA=C. IEEE Trans. Automatic Control. 1982, AC-27: 464-466
[20] Stewart, G. W.. Computing the CS-decomposition of a partitioned orthogonal matrix. Number.Math.. 1982, 40: 297-306
[21] Eurice de Souza and S. P. Bhattacharyya. Controliabiiity, observability and the solution of AX-XB=C. Linear Algebra and its Applications. 1981, 39: 167-188
[22] Paige, C. C. and Saunders, M. A.. Towards a generalized singular value decomposition. SIAM J. Numer. Anal.. 1981, 18: 398-405
[23] Wei F. S. Stiffness matrix correction from incomplete test data. AIAA J. 1980, 18(10): 1274-1275
[24] Berman A. Mass matrix correction using an incomplete set of measure modes. AIAA J. 1979, 17: 1147-1148
[25] Baruch M. Optimization procedure to correct stiffness and flexibility matrices using vibration tests. AIAA J. 1978, 16(11): 1208-1210
[26] Baruch M, Bar Itzhack IY. Optimal weighted orthogonalization of measured modes. AIAA J. 1978, 16(4): 345-351
[27] Jameson A. and Kreindler E. Inverse problem of linear Optimal control. SIAM J. Control. 1973, 11: 1-19
[28] A.Jameson. Solution of the equation AX+XB=C by inversion of an M×M or N×N matrix. SIAM J. Appl, Math.. 1968, 16: 1020-1023
[29] Brock I. E. Optimal matrices describing linear system. AIAA J. 1968, 6: 1292-1296
[30] 彭振赟,胡锡炎,张磊.双对称矩阵的一类反问题.计算数学.2005, 27(1):11-18
[31] 周富照,胡锡炎.反对称正交对称矩阵反问题.数学杂志.2005,25(2):179-184.
[32] 盛炎平,谢冬秀.一类对称次反对称矩阵反问题解存在的条件.计算数学2004,26(1):73-80
[33] 周富照,胡锡炎,张磊.对称正交反对称矩阵反问题.数学物理学报.2004,24A(5):543-550
[34] 戴华.对称正交对称矩阵反问题的最小二乘解.计算数学.2003,25(1):59-66
[35] 彭亚新,胡锡炎,张磊.矩阵方程A~TXA=B的对称正交对称解及其最佳逼近.高等学校计算数学学报.2003,25(4):372-377
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