关于正定矩阵的子式阵的正定性

研究正定矩阵的子矩阵,利用合同标准形分别给出了复正定矩阵的子式阵为复正定矩阵和实正定矩阵的子式阵为实正定矩阵的充分必要条件,其结果简单而实用.     

关于合成矩阵的正定性

利用标准形分别给出了复正定矩阵的合成矩阵为复正定矩阵和实正定矩阵的合成矩阵为实正定矩阵的充分必要条件,其结果简单而实用.     

互反判断矩阵一致性指标研究

首先给出了互反判断矩阵与一致性互反判断矩阵集之间距离的定义,基于此定义,提出了一个新的互反判断矩阵一致性指标,并给出了此一致性指标的度量方法。对于不满足此一致性指标的互反判断矩阵,提出了一个迭代算法来提高其一致性程度。得出了群体互反判断矩阵一致性指标的下界,为提出的一致性指标应用于群决策问题提供了理论基础。最后用数值例子说明了该迭代算法的可行性和有效性以及群决策中的相关结论。     

Hermite-广义反Hamilton矩阵的一类反问题

给定矩阵X和B,利用矩阵的广义奇异值分解,得到了矩阵方程X~HAX=B有Hermite-广义反Hamiton解的充分必要条件及有解时解的—般表达式.用S_E表示此矩阵方程的解集合,证明了S_E中存在唯一的矩阵(?),使得(?)与给定矩阵A的差的Frobenius范数最小,并且给出了矩阵(?)的表达式;同时也证明了S_E中存在唯一的矩阵A_o,使得A_o是此矩阵方程的极小Frobenius范数Hermite-广义反Hamilton解,并且给出了矩阵A_o的表达式.     

几类特殊矩阵的性质研究及应用

阐述了判断矩阵、度量矩阵及偏离矩阵的特征值相同性以及它们的特征向量的关系.用度量矩阵和判断矩阵的偏离矩阵作比较,这能够清楚地表明哪些元素对判断矩阵的一致性影响较大.     

Hermitian positive semidefinite matrices whose entries are 0 or 1 in Modulus

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it is similar to a direct sum of all I's matrices and a 0 matrix via a unitary monomial similarity. In particular, the only such nonsingular matrix is the identity matrix and the only such irreducible matrix is similar to an all l's matrix by means of a unitary diagonal similarity. Our results extend earlier results of Jain and Snyder for the case in which the nonzero entries (actually) equal 1. Our methods of proof, which reiy on the so called principal submatrix rank property, differ from the approach used by Jain and Snyder.     

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.     

Topological symmetry,background independence and matrix models

We illustrate a physical situation in which topological symmetry, its breakdown, space-time uncertainty principle, and background independence may play an important role in constructing and understanding matrix models. First, we show that the space-time uncertainty principle of string may be understood as a manifestation of the breakdown of the topological symmetry in the large N matrix model. Next, we construct a new type of matrix models which is a matrix model analog of the topological Chern-Simons and BF theories. It is of interest that these topological matrix models are not only completely independent of the background metric but also have nontrivial “p-brane” solutions as well as commuting classical space-time as the classical solutions. In this paper, we would like to point out some elementary and unsolved problems associated to the matrix models, whose resolution would lead to the more satisfying matrix model in future.     

The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms

The orthogonal Procrustes problem involves finding an orthogonal matrix which transforms one given matrix into another in the least-squares sense, and thus it requires the minimization of the Frobenius matrix norm. We consider, the solution of this problem for a family of orthogonally invariant norms which includes the Frobenius norm as a special case.     

The matrix sign function and computations in systems

The matrix sign function has several interesting properties which form the basis of new solution algorithms for problems which occur frequently in systems and control theory applications. Presented in this paper are new algorithms, based on the matrix sign function, for the solution of algebraic matrix Riccati equations, Lyapunov equations, coupled Riccati equations, spectral factorization, matrix square roots, pole assignment, and the algebraic eigenvalue-eigenvector problem. Examples of the application of each algorithm are also presented.     

Lattice orders on 2 × 2 triangular matrix algebras

We construct all the lattice orders on a 2 × 2 triangular matrix algebra over a totally ordered field that make it into a lattice-ordered algebra. It is shown that every lattice order in which the identity matrix is not positive may be obtained from a lattice order in which the identity matrix is positive.     

Numerically stable deflation of hessenberg and symmetric tridiagonal matrices

While numerically stable techniques have been available for deflating a fulln byn matrix, no satisfactory finite technique has been known which preserves Hessenberg form. We describe a new algorithm which explicitly deflates a Hessenberg matrix in floating point arithmetic by means of a sequence of plane rotations. When applied to a symmetric tridiagonal matrix, the deflated matrix is again symmetric tridiagonal. Repeated deflation can be used to find an orthogonal set of eigenvectors associated with any selection of eigenvalues of a symmetric tridiagonal matrix.     

Numerically Stable Generation of Correlation Matrices and Their Factors

Correlation matrices—symmetric positive semidefinite matrices with unit diagonal—are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigenvalues (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Givens rotations to introduce 1s on the diagonal. We give improved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, for example when the correlation matrix is nearly singular to working precision.     

AHP中不一致判断矩阵调整方法的改进

针对AHP中不一致性判断矩阵,提出了一种新的调整方法.通过分析诱导矩阵与判断矩阵之间的关系,用元素的几何平均值对矩阵中偏差最大的元素逐个修正,直到判断矩阵达到满意的一致性.该算法利用原始判断信息修正单个元素,提高了判断矩阵修正速度,简洁、实用.     

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.     

退化矩阵Liouville,Beta,Dirichlet分布

该文引进和讨论了退化矩阵Liouville分布，由此导出退化矩阵Beta分布、退化矩阵Dirichlet分布．推广了文献［1］关于退化Wishart分布和秩为1的退化矩阵Beta分布的结果。     

Characterization of symmetric M-matrices as resistive inverses

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do this we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector. We also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric M-matrices.     

On the use of dense matrix techniques within sparse simplex

In this paper we discuss some instances where dense matrix techniques can be utilized within a sparse simplex linear programming solver. The main emphasis is on the use of the Schur complement matrix as a part of the basis matrix representation. This approach enables to represent the basis matrix as an easily invertible sparse matrix and one or more dense Schur complement matrices. We describe our variant of this method which uses updating of the QR factorization of the Schur complement matrix. We also discuss some implementation issues of the LP software package which is based on this approach.     

不同直觉偏好信息下的多属性决策方法

研究了属性值为实数且决策者对属性的偏好信息以直觉判断矩阵或残缺直觉判断矩阵给出的模糊多属性决策问题.首先介绍了直觉判断矩阵、一致性直觉判断矩阵、残缺直觉判断矩阵、一致性残缺直觉判断矩阵等概念,而后分别考虑关于直觉判断矩阵和残缺直觉判断矩阵的多属性决策问题,接着建立了基于直觉判断矩阵和残缺直觉判断矩阵的多属性群决策模型,通过求解这些模型获得属性的权重.进而给出了不同直觉偏好信息下的多属性决策方法.最后通过一个例子说明了该方法的可行性和实用性.     

The direct product permuting matrices

A new matrix product is defined and its properties are investigated. The commutatuion matrix which flips a left direct product of two matrices into a right direct one is derived as a composition of two identity matrices. The communication matrix is a special case of the direct product permuting matrices defined in this paper which are matrix representations of the permutation operators on tensor spaces i e. the linear mappings which permute the order of the vectors in a direct product of them. Explicit expressions for these matrices are given. properties of the matrices are investigated and it is shown how these matrices, act on various representations of tensor spaces.