矩阵方程AXB=D的(R,S)-斜对称解
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摘要
矩阵方程AXB=D是教学、理论研究和工程实践中常见的一种矩阵方程.给出了AXB=D具有(R,S)-斜对称矩阵解的充分必要条件,及其解存在条件下全体解集合S_X的表达式.此外,还讨论了任意给定矩阵X在仿射子空间S_X中的最优近似解,并给出了最优解的显示表达式.
Let RSSC~(m×n) be the set of all m×n(R,S)-skew symmetric matrices.In this paper,a sufficient and necessary condition for the matrix equation AXB = D,where A∈C~(p×m),B∈C~(n×q),and D∈C~(p×q),to have a solution X∈RSSC~(m×n) is established.And if it exists,a representation of the solution set S_X is given.An optimal approximation between a given matrix X∈C~(m×n) and the affine subspace S_X is discussed, and an explicit formula for the unique optimal approximation solution is presented.
Let RSSC~(m×n) be the set of all m×n(R,S)-skew symmetric matrices.In this paper,a sufficient and necessary condition for the matrix equation AXB = D,where A∈C~(p×m),B∈C~(n×q),and D∈C~(p×q),to have a solution X∈RSSC~(m×n) is established.And if it exists,a representation of the solution set S_X is given.An optimal approximation between a given matrix X∈C~(m×n) and the affine subspace S_X is discussed, and an explicit formula for the unique optimal approximation solution is presented.
引文
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[2]Yuan Y X,Dai H..Generalized reflexive solutions of the matrix equation AXB = D and an associated optimal approximation problem[J].Computers and Mathematics with Applications, 2008,56:1643-1649.
[3]Khatri C G,Matra S K.Hermitian and nonnegative definite solutions of linear matrix equation [J].SIAM J App Math,1976,32:579-585.
[4]Wang J S,Chang X W.Symmetric solution of a linear matrix equation[J].Nanjing Univ J Math Biquarterly,1990,7:125-129.(in Chinease)
[5]Weaver J R.Centrosymmetric(cross-symmetric) matrices,their basic properties,eigenvalues, eigenvectors[J].Amer Math Monthly,1985,92:711-717.
[6]Peng Z Y,Hu X Y.The reflexive solutions of the matrix equation AXB = C[J].Comput Math Appl,2006,51:897-902.
[7]Trench W F.Minimization problems for(R,S)-symmetric and(R,5)-skew symmetric matrices[J]. Linear Algebra Appl,2004,389:23-31.
[8]Golub G H,Van Loan C F.Matrix Computations[M].Baltimore:The Johns Hopkins University Press,1983.
[9]Stewart G W.Computing the CS-decomposition of a partitioned orthogonal matrix[J].Numer Math,1982,40:297-306.
[10]Aubin J P.Applied Functional Analysis[M].New York:John Weiley,1979.