This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative ...This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .展开更多
Let A∈C<sup>m×n</sup>,set eigenvalues of matrix A with |λ<sub>1</sub> （A）|≥|λ<sub>2</sub>（A）|≥…≥|λ<sub>n</sub>（A）|,write A≥0 if A is a positive semid...Let A∈C<sup>m×n</sup>,set eigenvalues of matrix A with |λ<sub>1</sub> （A）|≥|λ<sub>2</sub>（A）|≥…≥|λ<sub>n</sub>（A）|,write A≥0 if A is a positive semidefinite Hermitian matrix, and denote∧<sub>k</sub> （A）=diag （λ<sub>1</sub>（A）,…,λ<sub>k</sub>（A））,∧<sub>(</sub>（n-k）.（A）=diag (λ<sub>k+1</sub>（A）,…,λ<sub>n</sub>（A）)for any k=1, 2,...,n if A≥0. Denote all n order unitary matrices by U<sup>n×n</sup>.Problem of equalities to hold in eigenvalue inequalities for products of matrices展开更多
Main resultsTheorem 1 Let A be symmetric positive semidefinite.Let (?) be a diagonally compen-sated reduced matrix of A and Let (?)=σI+(?)(σ】0) be a modiffication(Stieltjes) matrixof (?).Let the splitting (?)=M-(?)...Main resultsTheorem 1 Let A be symmetric positive semidefinite.Let (?) be a diagonally compen-sated reduced matrix of A and Let (?)=σI+(?)(σ】0) be a modiffication(Stieltjes) matrixof (?).Let the splitting (?)=M-(?) be regular and M=F-G be weak regular,where M andF are symmetric positive definite matrices.Then the resulting two-stage method corre-sponding to the diagonally compensated reduced splitting A=M-N and inner splitting M=F-G is convergent for any number μ≥1 of inner iterations.Furthermore,the展开更多
In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a u...In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.展开更多
Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm}，We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A...Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm}，We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best"approximate to λixi, i = 1, 2,..., m in Frobenius norm and (Ⅱ) the Y in set ( )which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem Ⅰ and Problem Ⅱ is given andthe general expression of solutions for Problem Ⅰ is derived. Some sufficient conditionsunder which Problem Ⅰ and Problem Ⅱ have an explicit solution is provided. A numer-ical algorithm of the solution for Problem Ⅱ has been presented.展开更多
To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a spe...To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a special form (e.g. the dissection form [11]). The main tool for deriving our methods is the diagonally compensated reduction (cf. [1]). The convergence of such methods is also discussed by using this tool. [WT5,5”HZ]展开更多
We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put...We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.展开更多
A parallel imaginary EBE (element-by-element )method for solving positive definite linear systems is presented. The EBE strategy is originally used as a sequential method[1.2], and later it is converted to a parallel...A parallel imaginary EBE (element-by-element )method for solving positive definite linear systems is presented. The EBE strategy is originally used as a sequential method[1.2], and later it is converted to a parallelmethod for solving finite element problem in solid mechanics[3]. The main contribution of this paper is to forma general parallel EBE method for the solution of anyPOsitive definite linear system through a so-called imaginary finite element technique. It is then POssible to use.finite elemental without finite element.展开更多
The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive...The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as . The numerical simulations illustrated our method have competitive advantage than other methods.展开更多
In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is pr...In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.展开更多
In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].
Based on our previously proposed Wigner operator in entangled form, we introduce the generalized Wigner operator for two entangled particles with different masses, which is expected to be positive-definite. This appro...Based on our previously proposed Wigner operator in entangled form, we introduce the generalized Wigner operator for two entangled particles with different masses, which is expected to be positive-definite. This approach is able to convert the generalized Wigner operator into a pure state so that the positivity can be ensured. The technique of integration within an ordered product of operators is used in the discussion.展开更多
Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square...Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.展开更多
For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 30...For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).展开更多
文摘This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .
基金Supported partly by National Natural Science Foundation of China
文摘Let A∈C<sup>m×n</sup>,set eigenvalues of matrix A with |λ<sub>1</sub> （A）|≥|λ<sub>2</sub>（A）|≥…≥|λ<sub>n</sub>（A）|,write A≥0 if A is a positive semidefinite Hermitian matrix, and denote∧<sub>k</sub> （A）=diag （λ<sub>1</sub>（A）,…,λ<sub>k</sub>（A））,∧<sub>(</sub>（n-k）.（A）=diag (λ<sub>k+1</sub>（A）,…,λ<sub>n</sub>（A）)for any k=1, 2,...,n if A≥0. Denote all n order unitary matrices by U<sup>n×n</sup>.Problem of equalities to hold in eigenvalue inequalities for products of matrices
文摘Main resultsTheorem 1 Let A be symmetric positive semidefinite.Let (?) be a diagonally compen-sated reduced matrix of A and Let (?)=σI+(?)(σ】0) be a modiffication(Stieltjes) matrixof (?).Let the splitting (?)=M-(?) be regular and M=F-G be weak regular,where M andF are symmetric positive definite matrices.Then the resulting two-stage method corre-sponding to the diagonally compensated reduced splitting A=M-N and inner splitting M=F-G is convergent for any number μ≥1 of inner iterations.Furthermore,the
基金This research is supported by the National Natural Science Foundation of China(No.11871444).
文摘In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.
基金Suported by National Nature Science Foundation of China
文摘Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm}，We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best"approximate to λixi, i = 1, 2,..., m in Frobenius norm and (Ⅱ) the Y in set ( )which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem Ⅰ and Problem Ⅱ is given andthe general expression of solutions for Problem Ⅰ is derived. Some sufficient conditionsunder which Problem Ⅰ and Problem Ⅱ have an explicit solution is provided. A numer-ical algorithm of the solution for Problem Ⅱ has been presented.
文摘To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a special form (e.g. the dissection form [11]). The main tool for deriving our methods is the diagonally compensated reduction (cf. [1]). The convergence of such methods is also discussed by using this tool. [WT5,5”HZ]
基金This work is supported by NSF of Shanxi province,20011041.
文摘We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.
文摘A parallel imaginary EBE (element-by-element )method for solving positive definite linear systems is presented. The EBE strategy is originally used as a sequential method[1.2], and later it is converted to a parallelmethod for solving finite element problem in solid mechanics[3]. The main contribution of this paper is to forma general parallel EBE method for the solution of anyPOsitive definite linear system through a so-called imaginary finite element technique. It is then POssible to use.finite elemental without finite element.
文摘The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as . The numerical simulations illustrated our method have competitive advantage than other methods.
文摘In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.
文摘We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.
文摘In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].
基金Project supported by the National Natural Science Foundation of China (Grant Nos.10874174 and 10947017/A05)the Key Programs Foundation of Ministry of Education of China (Grant No.210115)
文摘Based on our previously proposed Wigner operator in entangled form, we introduce the generalized Wigner operator for two entangled particles with different masses, which is expected to be positive-definite. This approach is able to convert the generalized Wigner operator into a pure state so that the positivity can be ensured. The technique of integration within an ordered product of operators is used in the discussion.
文摘Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.
文摘For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).
