A = (a[sub ij]) ∈ R[sup n×n] is termed bisymmetric matrix if a[sub ij] = a[sub ji] = a[sup n ? j + 1, n ? i + 1], i, j = 1, 2 ··· n. We denote the set of all n x n bisymmetric matrices by BSR[sup ...A = (a[sub ij]) ∈ R[sup n×n] is termed bisymmetric matrix if a[sub ij] = a[sub ji] = a[sup n ? j + 1, n ? i + 1], i, j = 1, 2 ··· n. We denote the set of all n x n bisymmetric matrices by BSR[sup n x n]. This paper is mainly concerned with solving the following two problems: Problem I. Given X, B ∈ R[sup n×m], find A ∈ P[sub n] such that AX = B, where P[sub n] = {A ∈ BSR[sup n×n]| x[sup T] Ax ≥ 0, ?x ∈ R[sup n]}. Problem II. Given A[sup *] ∈ R[sup n×n], find ? ∈ S[sub E] such that ||A[sup *] - ?||[sub F] = ... ||A[sup *] - A||[sub F] where || · ||[sub F] is Frobenius norm, and S[sub E] denotes the solution set of problem I. The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of S[sub E] has been given. For problem II the expression of the solution has been provided. [ABSTRACT FROM AUTHOR]展开更多
We define an m-involution to be a matrix K ∈ Cn×n for which Km -= I. In this article, we investigate the class Sm （A） of m-involutions that commute with a diagonalizable matrix A E Cn×n. A number of basic...We define an m-involution to be a matrix K ∈ Cn×n for which Km -= I. In this article, we investigate the class Sm （A） of m-involutions that commute with a diagonalizable matrix A E Cn×n. A number of basic properties of Sm （A） and its related subclass Sm （A, X） are given, where X is an eigenvector matrix of A. Among them, Sm （A） is shown to have a torsion group structure under matrix multiplication if A has distinct eigenvalues and has non-denumerable cardinality otherwise. The constructive definition of Sm （A, X） allows one to generate all m-involutions commuting with a matrix with distinct eigenvalues. Some related results are also given for the class S,, （A） of m-involutions that anti-commute with a matrix A ∈ Cnn×n.展开更多
In this paper, an iterative method is constructed to find the least-squares solutions of generalized Sylvester equation , where is real matrices group, and satisfies different linear constraint. By this iterative meth...In this paper, an iterative method is constructed to find the least-squares solutions of generalized Sylvester equation , where is real matrices group, and satisfies different linear constraint. By this iterative method, for any initial matrix group within a special constrained matrix set, a least squares solution group with satisfying different linear constraint can be obtained within finite iteration steps in the absence of round off errors, and the unique least norm least-squares solution can be obtained by choosing a special kind of initial matrix group. In addition, a minimization property of this iterative method is characterized. Finally, numerical experiments are reported to show the efficiency of the proposed method.展开更多
We study the symmetric positive semidefinite solution of the matrix equation AX_1A^T + BX_2B^T=C. where A is a given real m×n matrix. B is a given real m×p matrix, and C is a given real m×m matrix, with...We study the symmetric positive semidefinite solution of the matrix equation AX_1A^T + BX_2B^T=C. where A is a given real m×n matrix. B is a given real m×p matrix, and C is a given real m×m matrix, with m, n, p positive integers: and the bisymmetric positive semidefinite solution of the matrix equation D^T XD=C, where D is a given real n×m matrix. C is a given real m×m matrix, with m. n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions.展开更多
We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the ex...We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.展开更多
文摘A = (a[sub ij]) ∈ R[sup n×n] is termed bisymmetric matrix if a[sub ij] = a[sub ji] = a[sup n ? j + 1, n ? i + 1], i, j = 1, 2 ··· n. We denote the set of all n x n bisymmetric matrices by BSR[sup n x n]. This paper is mainly concerned with solving the following two problems: Problem I. Given X, B ∈ R[sup n×m], find A ∈ P[sub n] such that AX = B, where P[sub n] = {A ∈ BSR[sup n×n]| x[sup T] Ax ≥ 0, ?x ∈ R[sup n]}. Problem II. Given A[sup *] ∈ R[sup n×n], find ? ∈ S[sub E] such that ||A[sup *] - ?||[sub F] = ... ||A[sup *] - A||[sub F] where || · ||[sub F] is Frobenius norm, and S[sub E] denotes the solution set of problem I. The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of S[sub E] has been given. For problem II the expression of the solution has been provided. [ABSTRACT FROM AUTHOR]
文摘We define an m-involution to be a matrix K ∈ Cn×n for which Km -= I. In this article, we investigate the class Sm （A） of m-involutions that commute with a diagonalizable matrix A E Cn×n. A number of basic properties of Sm （A） and its related subclass Sm （A, X） are given, where X is an eigenvector matrix of A. Among them, Sm （A） is shown to have a torsion group structure under matrix multiplication if A has distinct eigenvalues and has non-denumerable cardinality otherwise. The constructive definition of Sm （A, X） allows one to generate all m-involutions commuting with a matrix with distinct eigenvalues. Some related results are also given for the class S,, （A） of m-involutions that anti-commute with a matrix A ∈ Cnn×n.
文摘In this paper, an iterative method is constructed to find the least-squares solutions of generalized Sylvester equation , where is real matrices group, and satisfies different linear constraint. By this iterative method, for any initial matrix group within a special constrained matrix set, a least squares solution group with satisfying different linear constraint can be obtained within finite iteration steps in the absence of round off errors, and the unique least norm least-squares solution can be obtained by choosing a special kind of initial matrix group. In addition, a minimization property of this iterative method is characterized. Finally, numerical experiments are reported to show the efficiency of the proposed method.
基金Subsidized by the Special Funds for Major State Basic Research Projects G1999032803
文摘We study the symmetric positive semidefinite solution of the matrix equation AX_1A^T + BX_2B^T=C. where A is a given real m×n matrix. B is a given real m×p matrix, and C is a given real m×m matrix, with m, n, p positive integers: and the bisymmetric positive semidefinite solution of the matrix equation D^T XD=C, where D is a given real n×m matrix. C is a given real m×m matrix, with m. n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions.
基金This research is supported by the Natural Science Foundation of China(No.0471085the Natural Science Foundation of Shanghai)the Development Foundation of Shanghai Educational Committee the Special Funds for Major Specialities of Shanghai Education Co
文摘We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.
