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
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 provide some new necessary and sufficient conditions for generalized diagonally dominant matrices and also obtain some criteria for nongeneralized dominant matrices.
On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial ...On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.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
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
One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so...One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The largest interval is given by which the diagonal entries of the inverse of tridiagonal M-matrices can be perturbed without losing the property of total nonnegativity. A numerical example is given to illustrate our findings.展开更多
Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact th...Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact that the manifold is ±l-flat is shown.Moreover,the divergence of two points on the manifold is given through dual potential functions.Furthermore,the optimal approximation of a point onto the submanifold is gotten.Finally,some simulations are given to illustrate our results.展开更多
In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system ...In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.展开更多
This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invar...This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invariant norms and positive semidefinite matrices, which generalize the result obtained by Bhatia and Kittaneh.展开更多
A real n × n symmetric matrix P is partially positive(PP) for a given index set I ? {1,..., n} if there exists a matrix V such that V(I, :) 0 and P = V VT. We give a characterization of PP-matrices. A semidefinit...A real n × n symmetric matrix P is partially positive(PP) for a given index set I ? {1,..., n} if there exists a matrix V such that V(I, :) 0 and P = V VT. We give a characterization of PP-matrices. A semidefinite algorithm is presented for checking whether a matrix is partially positive or not. Its properties are studied. A PP-decomposition of a matrix can also be obtained if it is partially positive.展开更多
In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an alg...In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an algorithm which is very similar to the primal-dual potential reduction algorithm of Huang and Kortanek [6] for linear programming. The complexity of the algorithm is either O(nlog(X0 · S0/ε) or O(nlog(X0· S0/ε) depends on the value of ρ in the primal-dual potential function, where X0 and S0 is the initial interior matrices of the positive semi-definite programming.展开更多
For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(...For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(A+B)≤sj((A⊕B)+φc(A,B))≤sj(A+|B1/2A1/2|)⊕(B+|A1/2B1/2|),where sj(X)denotes the j-th largest singular value of X andφc(A,B):=1/2((1+c)|B1/2A1/2|(1-c)A1/2B1/2(1-c)B1/2A1/2(1+c)|A1/2B1/2|).This result sharpens some known result.Meanwhile,some related results are established.展开更多
By presenting a counterexample, the author of paper （ZHAO Li-feng. J. Math. Res. Exposition, 2007, 27（4）： 949-954） declared that some assertions in papers of LU Yun-xia, ZHANG Shu-qing （J. Math. Res. Exposition,...By presenting a counterexample, the author of paper （ZHAO Li-feng. J. Math. Res. Exposition, 2007, 27（4）： 949-954） declared that some assertions in papers of LU Yun-xia, ZHANG Shu-qing （J. Math. Res. Exposition, 1999, 19（3）： 598-600）, HE Gan-tong （J. Math. Res. Exposition, 2002, 22（1）： 79-82） and YUAN Hui-ping （J. Math. Res. Exposition, 2001, 21（3）： 464-468） are wrong. In this note, we point out that the counterexample is wrong. Further discussion on these assertions and some related results are also given.展开更多
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.
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
文摘In this paper, we provide some new necessary and sufficient conditions for generalized diagonally dominant matrices and also obtain some criteria for nongeneralized dominant matrices.
文摘On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.
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One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The largest interval is given by which the diagonal entries of the inverse of tridiagonal M-matrices can be perturbed without losing the property of total nonnegativity. A numerical example is given to illustrate our findings.
基金Supported by Natural Science Foundations of China(Grant No.61179031 and 61401058)
文摘Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact that the manifold is ±l-flat is shown.Moreover,the divergence of two points on the manifold is given through dual potential functions.Furthermore,the optimal approximation of a point onto the submanifold is gotten.Finally,some simulations are given to illustrate our results.
文摘In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.
基金Supported by the Scientific Research Project of Chongqing Three Gorges University(11QN-21)
文摘This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invariant norms and positive semidefinite matrices, which generalize the result obtained by Bhatia and Kittaneh.
基金supported by National Natural Science Foundation of China(Grant No.11171217)
文摘A real n × n symmetric matrix P is partially positive(PP) for a given index set I ? {1,..., n} if there exists a matrix V such that V(I, :) 0 and P = V VT. We give a characterization of PP-matrices. A semidefinite algorithm is presented for checking whether a matrix is partially positive or not. Its properties are studied. A PP-decomposition of a matrix can also be obtained if it is partially positive.
基金This research was partially supported by a fund from Chinese Academy of Science,and a fund from the Personal Department of the State Council.It is also sponsored by scientific research foundation for returned overseas Chinese Scholars,State Education
文摘In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an algorithm which is very similar to the primal-dual potential reduction algorithm of Huang and Kortanek [6] for linear programming. The complexity of the algorithm is either O(nlog(X0 · S0/ε) or O(nlog(X0· S0/ε) depends on the value of ρ in the primal-dual potential function, where X0 and S0 is the initial interior matrices of the positive semi-definite programming.
基金Supported by the Natural Science Foundation of Anhui Province(1708085QA05)the Natural Science Foundation of Anhui Higher Education Institutions of China(KJ2019A0588,KJ2020ZD008)。
文摘For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(A+B)≤sj((A⊕B)+φc(A,B))≤sj(A+|B1/2A1/2|)⊕(B+|A1/2B1/2|),where sj(X)denotes the j-th largest singular value of X andφc(A,B):=1/2((1+c)|B1/2A1/2|(1-c)A1/2B1/2(1-c)B1/2A1/2(1+c)|A1/2B1/2|).This result sharpens some known result.Meanwhile,some related results are established.
基金Supported by the Natural Science Foundation of Science and Technology Office of Guizhou Province (Grant No. J[2006]2002)
文摘By presenting a counterexample, the author of paper （ZHAO Li-feng. J. Math. Res. Exposition, 2007, 27（4）： 949-954） declared that some assertions in papers of LU Yun-xia, ZHANG Shu-qing （J. Math. Res. Exposition, 1999, 19（3）： 598-600）, HE Gan-tong （J. Math. Res. Exposition, 2002, 22（1）： 79-82） and YUAN Hui-ping （J. Math. Res. Exposition, 2001, 21（3）： 464-468） are wrong. In this note, we point out that the counterexample is wrong. Further discussion on these assertions and some related results are also given.
