By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain a general expression of the least-squares solutions of the matrix equation ATXA = D under the restriction that the solution matrix ...By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain a general expression of the least-squares solutions of the matrix equation ATXA = D under the restriction that the solution matrix ∈ Rn×n is bisymmetric, where A ∈Rn×m and D ∈Rm×m are given matrices.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 study the Hermitian positive definite solutions of the matrix equation X +A*X^-qA = I with q > 0. Some properties of the solutions and the basic fixed point iterations for the equation are also discussed in some...We study the Hermitian positive definite solutions of the matrix equation X +A*X^-qA = I with q > 0. Some properties of the solutions and the basic fixed point iterations for the equation are also discussed in some detail. Some of results in [Linear Algebra Appl., 279 (1998), 303-316], [Linear Algebra Appl., 326 (2001),27-44] and [Linear Algebra Appl. 372 (2003), 295-304] are extended.展开更多
Least-squares solution of AXB = D with respect to symmetric positive semidefinite matrix X is considered. By making use of the generalized singular value decomposition, we derive general analytic formulas, and present...Least-squares solution of AXB = D with respect to symmetric positive semidefinite matrix X is considered. By making use of the generalized singular value decomposition, we derive general analytic formulas, and present necessary and sufficient conditions for guaranteeing the existence of the solution. By applying MATLAB 5.2, we give some numerical examples to show the feasibility and accuracy of this construction technique in the finite precision arithmetic.展开更多
We consider a profound problem of two-point resistance in the resistor network with a null resistor edge and an arbitrary boundary,which has not been solved before because the Green＇s function technique and the Lapla...We consider a profound problem of two-point resistance in the resistor network with a null resistor edge and an arbitrary boundary,which has not been solved before because the Green＇s function technique and the Laplacian matrix approach are invalid in this case.Looking for the exact solutions of resistance is important but difficult in the case of the arbitrary boundary since the boundary is a wall or trap which affects the behavior of a finite network.In this paper,we give a general resistance formula that is composed of a single summation by using the recursion-transform method.Meanwhile,several interesting results are derived by the general formula.Further,the current distribution is given explicitly as a byproduct of the method.展开更多
Few studies of wave propagation in layered saturated soils have been reported in the literature.In this paper,a general solution of the equation of wave motion in saturated soils,based on one kind of practical Blot...Few studies of wave propagation in layered saturated soils have been reported in the literature.In this paper,a general solution of the equation of wave motion in saturated soils,based on one kind of practical Blot's equation, was deduced by introducing wave potentials.Then exact dynamic-stiffness matrices for a poroelastic soil layer and half- space were derived,which extended Wolf's theory for an elastic layered site to the case of poroelasticity,thus resolving a fundamental problem in the field of wave propagation and soil-structure interaction in a poroelastic layered soil site.By using the integral transform method,Green's functions of horizontal and vertical uniformly distributed loads in a poroelastic layered soil site were given.Finally,the theory was verified by numerical examples and dynamic responses by comparing three different soil sites.This study has the following advantages:all parameters in the dynamic-stiffness matrices have explicitly physical meanings and the thickness of the sub-layers does not affect the precision of the calculation which is very convenient for engineering applications.The present theory can degenerate into Wolf's theory and yields numerical results approaching those for an ideal elastic layered site when porosity tends to zero.展开更多
In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ R...In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ Rm×n, ATX - XTA = D} find X ∈ S2 such that ||X|| = min. Problem II. Given A ∈ Rm×n,B B∈Rn×p, D ∈Rm×p. Let L1 = {X: X ∈ SRn×n, AXB = D} find X ∈ L1 such that ||X|| = min. Problem III. Given A ∈ Rm×n,B ∈ Rp×q, C ∈ Rm×q, G ∈ Rl×n, H ∈ Rp×t,D ∈ Rl × t. Let L2 = {X: X ∈ Rn×p, AXB = C, GXH = D} find X ∈ L2 such that ||X|| = min. Using singularvalue and canonical correlation decompositions, the necessary and sufficiellt conditions, under which S2, L1 and L2 are nonempty, are studied. The expressions for the solutions of Problems I, II and III are given.展开更多
文摘By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain a general expression of the least-squares solutions of the matrix equation ATXA = D under the restriction that the solution matrix ∈ Rn×n is bisymmetric, where A ∈Rn×m and D ∈Rm×m are given matrices.
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.
文摘We study the Hermitian positive definite solutions of the matrix equation X +A*X^-qA = I with q > 0. Some properties of the solutions and the basic fixed point iterations for the equation are also discussed in some detail. Some of results in [Linear Algebra Appl., 279 (1998), 303-316], [Linear Algebra Appl., 326 (2001),27-44] and [Linear Algebra Appl. 372 (2003), 295-304] are extended.
基金Subsidized by The Special Funds For Major State Basic Research Project G1999032803.
文摘Least-squares solution of AXB = D with respect to symmetric positive semidefinite matrix X is considered. By making use of the generalized singular value decomposition, we derive general analytic formulas, and present necessary and sufficient conditions for guaranteeing the existence of the solution. By applying MATLAB 5.2, we give some numerical examples to show the feasibility and accuracy of this construction technique in the finite precision arithmetic.
文摘We consider a profound problem of two-point resistance in the resistor network with a null resistor edge and an arbitrary boundary,which has not been solved before because the Green＇s function technique and the Laplacian matrix approach are invalid in this case.Looking for the exact solutions of resistance is important but difficult in the case of the arbitrary boundary since the boundary is a wall or trap which affects the behavior of a finite network.In this paper,we give a general resistance formula that is composed of a single summation by using the recursion-transform method.Meanwhile,several interesting results are derived by the general formula.Further,the current distribution is given explicitly as a byproduct of the method.
基金National Natural Science Foundation of China Under Grant No.50378063
文摘Few studies of wave propagation in layered saturated soils have been reported in the literature.In this paper,a general solution of the equation of wave motion in saturated soils,based on one kind of practical Blot's equation, was deduced by introducing wave potentials.Then exact dynamic-stiffness matrices for a poroelastic soil layer and half- space were derived,which extended Wolf's theory for an elastic layered site to the case of poroelasticity,thus resolving a fundamental problem in the field of wave propagation and soil-structure interaction in a poroelastic layered soil site.By using the integral transform method,Green's functions of horizontal and vertical uniformly distributed loads in a poroelastic layered soil site were given.Finally,the theory was verified by numerical examples and dynamic responses by comparing three different soil sites.This study has the following advantages:all parameters in the dynamic-stiffness matrices have explicitly physical meanings and the thickness of the sub-layers does not affect the precision of the calculation which is very convenient for engineering applications.The present theory can degenerate into Wolf's theory and yields numerical results approaching those for an ideal elastic layered site when porosity tends to zero.
文摘In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ Rm×n, ATX - XTA = D} find X ∈ S2 such that ||X|| = min. Problem II. Given A ∈ Rm×n,B B∈Rn×p, D ∈Rm×p. Let L1 = {X: X ∈ SRn×n, AXB = D} find X ∈ L1 such that ||X|| = min. Problem III. Given A ∈ Rm×n,B ∈ Rp×q, C ∈ Rm×q, G ∈ Rl×n, H ∈ Rp×t,D ∈ Rl × t. Let L2 = {X: X ∈ Rn×p, AXB = C, GXH = D} find X ∈ L2 such that ||X|| = min. Using singularvalue and canonical correlation decompositions, the necessary and sufficiellt conditions, under which S2, L1 and L2 are nonempty, are studied. The expressions for the solutions of Problems I, II and III are given.
