Let R be an associative ring with unity 1. The existence of the Moore-Penrose inverses of block matrices overR is investigated and the sufficient ad necessary conditions for such existence are obtained. Furthermore, ...Let R be an associative ring with unity 1. The existence of the Moore-Penrose inverses of block matrices overR is investigated and the sufficient ad necessary conditions for such existence are obtained. Furthermore, the representation of the Moore-Penrose inverse of M=[0 A C B]is given under the condition of EBF - 0, where E - I - CCT and F - I -AfA. This result generalizes the representation of the Moore-Penrose inverse of the companion matrix M =[0 a In b]due to Pedro Patricio. As for applications, some examples are given to illustrate the obtained results.展开更多
This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexi...This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexity of the method is analyzed and a numerical example is included. A variant of the algorithm with lower computational complexity is also proposed. Both algorithms are tested on randomly generated matrices. Numerical performance confirms our theoretic results.展开更多
In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂...In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.展开更多
In this paper, the reverse order law for the Moore-Penrose inverse of closed linear operators with closed range is investigated by virtue of the Norm-preserving extension of the bounded linear operators. The results g...In this paper, the reverse order law for the Moore-Penrose inverse of closed linear operators with closed range is investigated by virtue of the Norm-preserving extension of the bounded linear operators. The results generalize some results obtained by S Izumino in [12].展开更多
The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-ci...The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.展开更多
Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).
This paper establishes some perturbation analysis for the tensor inverse,the tensor Moore-Penrose inverse,and the tensor system based on the t-product.In the settings of structured perturbations,we generalize the Sher...This paper establishes some perturbation analysis for the tensor inverse,the tensor Moore-Penrose inverse,and the tensor system based on the t-product.In the settings of structured perturbations,we generalize the Sherman-Morrison-Woodbury(SMW)formula to the t-product tensor scenarios.The SMW formula can be used to perform the sensitivity analy-sis for a multilinear system of equations.展开更多
In this paper, we study the existence of solutions for the semilinear equation , where A is a , , and is a nonlinear continuous function. Assuming that the Moore-Penrose inverse AT(AAT)-1?exists (A denotes the transpo...In this paper, we study the existence of solutions for the semilinear equation , where A is a , , and is a nonlinear continuous function. Assuming that the Moore-Penrose inverse AT(AAT)-1?exists (A denotes the transposed matrix of A) which is true whenever the determinant of the matrix AAT is different than zero, and the following condition on the nonlinear term satisfied . We prove that the semilinear equation has solutions for all. Moreover, these solutions can be found from the following fixed point relation .展开更多
A class of matrix inverse problems minimizing ‖A-‖ F on the linear manifold l A=｛A∈R n×m ｜‖AX-B‖ F=min｝ is considered. The perturbation analysis of the solution to these problems is carried out. Th...A class of matrix inverse problems minimizing ‖A-‖ F on the linear manifold l A=｛A∈R n×m ｜‖AX-B‖ F=min｝ is considered. The perturbation analysis of the solution to these problems is carried out. The perturbation upper bounds of the solution are given for both the consistent and inconsistent cases. The obtained preturbation upper bounds are with respect to the distance from the perturbed solution to the unperturbed manifold.展开更多
The derivation of a diagonally loaded sample-matrix inversion （LSMI） algorithm on the busis of inverse matrix recursion （i.e.LSMI-IMR algorithm） is conducted by reconstructing the recursive formulation of covarian...The derivation of a diagonally loaded sample-matrix inversion （LSMI） algorithm on the busis of inverse matrix recursion （i.e.LSMI-IMR algorithm） is conducted by reconstructing the recursive formulation of covariance matrix. For the new algorithm, diagonal loading is by setting initial inverse matrix without any addition of computation. In addition, a corresponding improved recursive algorithm is presented, which is low computational complexity. This eliminates the complex multiplications of the scalar coefficient and updating matrix, resulting in significant computational savings. Simulations show that the LSMI-IMR algorithm is valid.展开更多
The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise w...The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.展开更多
The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which s...The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which some special cases are discussed.展开更多
A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. T...A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.展开更多
An effective numerical algorithm for computing the determinant of a pentadiagonal Toeplitz matrix has been proposed by Xiao-Guang Lv and others [1]. The complexity of the algorithm is (9n + 3). In this paper, a new al...An effective numerical algorithm for computing the determinant of a pentadiagonal Toeplitz matrix has been proposed by Xiao-Guang Lv and others [1]. The complexity of the algorithm is (9n + 3). In this paper, a new algorithm with the cost of (4n + 6) is presented to compute the determinant of a pentadiagonal Toeplitz matrix. The inverse of a pentadiagonal Toeplitz matrix is also considered.展开更多
This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form,...This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form, then exporting flat quaternions base in two matrix form. It establishes serial 6R manipulator kinematic equations in the form of quaternion matrix. Then five variables are eliminated through linear elimination and application of lexicographic Groebner base. Thus, upper bound of the degree of the equation is determined, which is 16. In this way, a 16-degree equation with single variable is obtained without any extraneous root. This is the first time that quaternion matrix modeling has been used in 6R robot inverse kinematics analysis.展开更多
A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical...A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical formation design. In order to study the elliptical control strategy and perturbation effects, it is necessary to derive the inverse transformation of the relative state transition matrix based on relative orbit elements. Poisson bracket theory is used to obtain the linear transformations between the two representations: the relative orbit elements and the geocentric orbital frame. In this paper, the details of these transformations are presented.展开更多
In this study,based on an iterative method to solve nonlinear equations,a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed.Nu...In this study,based on an iterative method to solve nonlinear equations,a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed.Numerical compar-isons of the proposed method with other methods show that the average number of iterations,number of the Einstein products,and CPU time of our method are considerably less than other methods.In some applications,partial and fractional differential equations that lead to sparse matrices are considered as prototypes.We use the iterates obtained by the method as a preconditioner,based on tensor form to solve the multilinear system A∗N X=B.Finally,several practical numerical examples are also given to display the accuracy and efficiency of the new method.The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.展开更多
Censider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and sufficient conditions for the solvability, as well as the general solution are obtained. Th...Censider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and sufficient conditions for the solvability, as well as the general solution are obtained. The best approximate solution by the above solution set is given. Thus the open problem in [1] is solved.展开更多
The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions fo...The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.展开更多
基金The National Natural Science Foundation of China(No.11371089)the Natural Science Foundation of Jiangsu Province(No.BK20141327)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(No.20120092110020)the Natural Science Foundation of Jiangsu Higher Education Institutions of China(No.15KJB110021)
文摘Let R be an associative ring with unity 1. The existence of the Moore-Penrose inverses of block matrices overR is investigated and the sufficient ad necessary conditions for such existence are obtained. Furthermore, the representation of the Moore-Penrose inverse of M=[0 A C B]is given under the condition of EBF - 0, where E - I - CCT and F - I -AfA. This result generalizes the representation of the Moore-Penrose inverse of the companion matrix M =[0 a In b]due to Pedro Patricio. As for applications, some examples are given to illustrate the obtained results.
文摘This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexity of the method is analyzed and a numerical example is included. A variant of the algorithm with lower computational complexity is also proposed. Both algorithms are tested on randomly generated matrices. Numerical performance confirms our theoretic results.
基金Supported by the Nature Science Foundation of China(11471091 and 11401143)
文摘In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.
文摘In this paper, the reverse order law for the Moore-Penrose inverse of closed linear operators with closed range is investigated by virtue of the Norm-preserving extension of the bounded linear operators. The results generalize some results obtained by S Izumino in [12].
文摘The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.
文摘Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).
基金supported by the National Natural Science Foundation of China under grant number 11801534.
文摘This paper establishes some perturbation analysis for the tensor inverse,the tensor Moore-Penrose inverse,and the tensor system based on the t-product.In the settings of structured perturbations,we generalize the Sherman-Morrison-Woodbury(SMW)formula to the t-product tensor scenarios.The SMW formula can be used to perform the sensitivity analy-sis for a multilinear system of equations.
文摘In this paper, we study the existence of solutions for the semilinear equation , where A is a , , and is a nonlinear continuous function. Assuming that the Moore-Penrose inverse AT(AAT)-1?exists (A denotes the transposed matrix of A) which is true whenever the determinant of the matrix AAT is different than zero, and the following condition on the nonlinear term satisfied . We prove that the semilinear equation has solutions for all. Moreover, these solutions can be found from the following fixed point relation .
文摘A class of matrix inverse problems minimizing ‖A-‖ F on the linear manifold l A=｛A∈R n×m ｜‖AX-B‖ F=min｝ is considered. The perturbation analysis of the solution to these problems is carried out. The perturbation upper bounds of the solution are given for both the consistent and inconsistent cases. The obtained preturbation upper bounds are with respect to the distance from the perturbed solution to the unperturbed manifold.
文摘The derivation of a diagonally loaded sample-matrix inversion （LSMI） algorithm on the busis of inverse matrix recursion （i.e.LSMI-IMR algorithm） is conducted by reconstructing the recursive formulation of covariance matrix. For the new algorithm, diagonal loading is by setting initial inverse matrix without any addition of computation. In addition, a corresponding improved recursive algorithm is presented, which is low computational complexity. This eliminates the complex multiplications of the scalar coefficient and updating matrix, resulting in significant computational savings. Simulations show that the LSMI-IMR algorithm is valid.
文摘The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.
文摘The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which some special cases are discussed.
文摘A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.
文摘An effective numerical algorithm for computing the determinant of a pentadiagonal Toeplitz matrix has been proposed by Xiao-Guang Lv and others [1]. The complexity of the algorithm is (9n + 3). In this paper, a new algorithm with the cost of (4n + 6) is presented to compute the determinant of a pentadiagonal Toeplitz matrix. The inverse of a pentadiagonal Toeplitz matrix is also considered.
文摘This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form, then exporting flat quaternions base in two matrix form. It establishes serial 6R manipulator kinematic equations in the form of quaternion matrix. Then five variables are eliminated through linear elimination and application of lexicographic Groebner base. Thus, upper bound of the degree of the equation is determined, which is 16. In this way, a 16-degree equation with single variable is obtained without any extraneous root. This is the first time that quaternion matrix modeling has been used in 6R robot inverse kinematics analysis.
文摘A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical formation design. In order to study the elliptical control strategy and perturbation effects, it is necessary to derive the inverse transformation of the relative state transition matrix based on relative orbit elements. Poisson bracket theory is used to obtain the linear transformations between the two representations: the relative orbit elements and the geocentric orbital frame. In this paper, the details of these transformations are presented.
基金funded by Iran National Science Foundation(INSF)under Project No.4013447.
文摘In this study,based on an iterative method to solve nonlinear equations,a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed.Numerical compar-isons of the proposed method with other methods show that the average number of iterations,number of the Einstein products,and CPU time of our method are considerably less than other methods.In some applications,partial and fractional differential equations that lead to sparse matrices are considered as prototypes.We use the iterates obtained by the method as a preconditioner,based on tensor form to solve the multilinear system A∗N X=B.Finally,several practical numerical examples are also given to display the accuracy and efficiency of the new method.The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.
文摘Censider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and sufficient conditions for the solvability, as well as the general solution are obtained. The best approximate solution by the above solution set is given. Thus the open problem in [1] is solved.
文摘The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.
