Let A ∈ B（X） and B ∈ B（Y）, Me be an operator on Banach space X ＋ Y given by Mc =（A C 0 B）A generalized Drazin spectrum defined by σgD（T） = {λ∈C ： T-λI is not generalized Drazin invertible} is considere...Let A ∈ B（X） and B ∈ B（Y）, Me be an operator on Banach space X ＋ Y given by Mc =（A C 0 B）A generalized Drazin spectrum defined by σgD（T） = {λ∈C ： T-λI is not generalized Drazin invertible} is considered in this paper. It is shown that展开更多
This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs...This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.展开更多
Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necess...Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necessary and sufficient conditions for solvability of the problem and an expression for general solution of the matrix equation AX=B.In addition,we also obtain the expression for the solution of a relevant optimal approximate problem.展开更多
Generalize reflexive matrices are a special class of matrices ?that have the relation where? and ?are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in man...Generalize reflexive matrices are a special class of matrices ?that have the relation where? and ?are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in many scientific and engineering applications. They are also a generalization of centrosymmetric matrices and reflexive matrices. The main purpose of this paper is to present block decomposition schemes for generalized reflexive matrices of various types and to obtain their decomposed explicit block-diagonal structures. The decompositions make use of unitary equivalence transformations and, therefore, preserve the singular values of the matrices. They lead to more efficient sequential computations and at the same time induce large-grain parallelism as a by-product, making themselves computationally attractive for large-scale applications. A numerical example is employed to show the usefulness of the developed explicit decompositions for decoupling linear least-square problems whose coefficient matrices are of this class into smaller and independent subproblems.展开更多
@1 Definition 1 Let A=(α<sub>ij</sub>)∈C<sup>n×n</sup>,B=(b<sub>ij</sub>)∈C<sup>n×n</sup>,is nonsingular.The generalizedsingular values of A（relative to B...@1 Definition 1 Let A=(α<sub>ij</sub>)∈C<sup>n×n</sup>,B=(b<sub>ij</sub>)∈C<sup>n×n</sup>,is nonsingular.The generalizedsingular values of A（relative to B）are following determinate nonnegative real numberswhen ||·||<sub>2</sub> denotes the Euclid vector norm,〈n〉={1,2,…,n}.Definition 2 Let A,B∈C<sup>n×n</sup>,if there exist λ∈C and x∈C<sup>n</sup>\{0}。展开更多
The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove th...The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.展开更多
Stair matrices and their generalizations are introduced. The definitions and some properties of the matrices were first given by Lu Hao. This class of matrices provide bases of matrix splittings for iterative methods....Stair matrices and their generalizations are introduced. The definitions and some properties of the matrices were first given by Lu Hao. This class of matrices provide bases of matrix splittings for iterative methods. The remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the accelerated overrelaxation method （GAOR） is introduced. Some theories of the AOR method are extended to the generalized method to include a wide class of matrices. The convergence of the new method is derived for Hermitian positive definite matrices. Finally, some examples are given in order to show the superiority of the new method.展开更多
Let fs,t（m, n） be the number of （0, 1） - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones （sm = nt）. How to determine fs,t（m,n）？ As R. P. Stanley has obser...Let fs,t（m, n） be the number of （0, 1） - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones （sm = nt）. How to determine fs,t（m,n）？ As R. P. Stanley has observed （Enumerative Combinatorics I （1997）, Example 1.1.3）, the determination of fs,t（m, n） is an unsolved problem, except for very small s, t. In this paper the closed formulas for f2,2（n, n）, f3,2（m, n）, f4,2（m, n） are given. And recursion formulas and generating functions are discussed.展开更多
Let A be a Banach algebra with unit e and a,b,c∈A,Mc=(a c 0 b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of is d...Let A be a Banach algebra with unit e and a,b,c∈A,Mc=(a c 0 b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of is defined byσ_(gD)(α)={λ∈C:α-λe is not generalized Drazin invertible}.It is shown thatσ_(gD)(a)∪σ_(gD)(b)=σ_(gD)(M_(C))∪W_(2),where W_(g) is a union of certain holes σ_(gD) and W_(g)■σ_(gD)(a)∩σ_(gD)(b),or more finely.In addition,some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.展开更多
Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction for many mixed orthogonal arrays. But t...Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction for many mixed orthogonal arrays. But there are also orthogonal arrays which cannot be obtained by the usual difference matrices, such as mixed orthogonal arrays of run size 60. In order to construct these mixed orthogonal arrays, a class of special so-called generalized difference matrices were discovered by Zhang (1989,1990,1993,2006) from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented and proved. As an application, a lot of new orthogonal arrays of run size 60 have been constructed.展开更多
In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the ...In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.展开更多
We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-defi...We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.展开更多
Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I ex...Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.展开更多
The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ...The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ C^n×n and nonempty α （n） = {1,2,... ,n} are proposed such that A is an invertible H-matrix if A（α） and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A E HM and the subset α C （n） such that the Schur complement matrix A/α∈ HI^n-｜α｜ or A/α ∈ Hn-｜α｜^M or A/α ∈ H^n-｜ α｜^S.展开更多
基金Supported by the National Natural Science Foundation of China(11301077,1117131,11171066 and11226113)Foundation of the Education Department of Fujian Province(JA12074)the Natural ScienceFoundation of Fujian Province(2012J05003)
文摘Let A ∈ B（X） and B ∈ B（Y）, Me be an operator on Banach space X ＋ Y given by Mc =（A C 0 B）A generalized Drazin spectrum defined by σgD（T） = {λ∈C ： T-λI is not generalized Drazin invertible} is considered in this paper. It is shown that
文摘This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.
基金This research was supported by the NSF of China under grant number 10571047.
文摘Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necessary and sufficient conditions for solvability of the problem and an expression for general solution of the matrix equation AX=B.In addition,we also obtain the expression for the solution of a relevant optimal approximate problem.
文摘Generalize reflexive matrices are a special class of matrices ?that have the relation where? and ?are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in many scientific and engineering applications. They are also a generalization of centrosymmetric matrices and reflexive matrices. The main purpose of this paper is to present block decomposition schemes for generalized reflexive matrices of various types and to obtain their decomposed explicit block-diagonal structures. The decompositions make use of unitary equivalence transformations and, therefore, preserve the singular values of the matrices. They lead to more efficient sequential computations and at the same time induce large-grain parallelism as a by-product, making themselves computationally attractive for large-scale applications. A numerical example is employed to show the usefulness of the developed explicit decompositions for decoupling linear least-square problems whose coefficient matrices are of this class into smaller and independent subproblems.
文摘@1 Definition 1 Let A=(α<sub>ij</sub>)∈C<sup>n×n</sup>,B=(b<sub>ij</sub>)∈C<sup>n×n</sup>,is nonsingular.The generalizedsingular values of A（relative to B）are following determinate nonnegative real numberswhen ||·||<sub>2</sub> denotes the Euclid vector norm,〈n〉={1,2,…,n}.Definition 2 Let A,B∈C<sup>n×n</sup>,if there exist λ∈C and x∈C<sup>n</sup>\{0}。
文摘The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.
基金Project supported by the Natural Science Foundation of Liaoning Province of China (No.20022021)
文摘Stair matrices and their generalizations are introduced. The definitions and some properties of the matrices were first given by Lu Hao. This class of matrices provide bases of matrix splittings for iterative methods. The remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the accelerated overrelaxation method （GAOR） is introduced. Some theories of the AOR method are extended to the generalized method to include a wide class of matrices. The convergence of the new method is derived for Hermitian positive definite matrices. Finally, some examples are given in order to show the superiority of the new method.
文摘Let fs,t（m, n） be the number of （0, 1） - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones （sm = nt）. How to determine fs,t（m,n）？ As R. P. Stanley has observed （Enumerative Combinatorics I （1997）, Example 1.1.3）, the determination of fs,t（m, n） is an unsolved problem, except for very small s, t. In this paper the closed formulas for f2,2（n, n）, f3,2（m, n）, f4,2（m, n） are given. And recursion formulas and generating functions are discussed.
文摘Let A be a Banach algebra with unit e and a,b,c∈A,Mc=(a c 0 b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of is defined byσ_(gD)(α)={λ∈C:α-λe is not generalized Drazin invertible}.It is shown thatσ_(gD)(a)∪σ_(gD)(b)=σ_(gD)(M_(C))∪W_(2),where W_(g) is a union of certain holes σ_(gD) and W_(g)■σ_(gD)(a)∩σ_(gD)(b),or more finely.In addition,some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.
基金supported by Visiting Scholar Foundation of Key Lab in University and by National Natural Science Foundation of China (Grant No. 10571045)Specialized Research Fund for the Doctoral Program of Higher Education of Ministry of Education of China (Grant No. 44k55050)
文摘Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction for many mixed orthogonal arrays. But there are also orthogonal arrays which cannot be obtained by the usual difference matrices, such as mixed orthogonal arrays of run size 60. In order to construct these mixed orthogonal arrays, a class of special so-called generalized difference matrices were discovered by Zhang (1989,1990,1993,2006) from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented and proved. As an application, a lot of new orthogonal arrays of run size 60 have been constructed.
基金Supported by the Key Discipline Construction Project of Tianshui Normal University
文摘In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.
文摘We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.
文摘Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.
基金Acknowledgements This work was supported in part by the Science Foundation of the Education Department of Shaanxi Province of China （No. 2013JK0593）, the Scientific Research Foundation of Xi＇an Polytechnic University （No. BS1014）, the China Postdoctoral Science Foundation （No. 20110491668）, and the National Natural Science Foundations of China （Grant Nos. 11201362, 11271297, 11101325, 11171270）.
文摘The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ C^n×n and nonempty α （n） = {1,2,... ,n} are proposed such that A is an invertible H-matrix if A（α） and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A E HM and the subset α C （n） such that the Schur complement matrix A/α∈ HI^n-｜α｜ or A/α ∈ Hn-｜α｜^M or A/α ∈ H^n-｜ α｜^S.
