由混合谱数据构造矩阵

The eigenvalues and singular values are two of the most distinguished characteristics in a square matrix. Weyl has proved the majorization between them. Horn has proved its inverse, i.e. there exists a matrix with prescribed eigenvalues and singular values. This paper presents a direct transform method which shows the matrix can be upper triangular with its diagonal elements in any order. There exists a real-valued matrix with prescribed complex-conjugate eigenvalues and singular values. Construction of matrices with mixed data is also considered.     

基于矩阵幂运算的重特征值存在性定理

对于判断矩阵重特征值的存在性问题,运用“若λ是矩阵A的特征值,则入“是Ak的特征值”这一性质,通过矩阵的迹与特征值的关系,得到了实数域上矩阵重特征值的存在性定理并给出了证明．定理实现了“由矩阵幂运算来判断矩阵重特征值的存在性”这样一个计算过程,对讨论矩阵特征值问题具有一定的启示意义．     

Construction of band matrices from spectral data

A stable numerical algorithm is presented to generate a symmetric p-band matrix from the given eigenvalues of the p greatest leading submatrices. The algorithm consists of two parts. First a matrix with the given spectral data is constructed; then this matrix is transformed into a p-band matrix leaving invariant the eigenvalues of the p greatest leading submatrices.     

Computation of roots of real and complex matrices

An algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix. All required computations are in the real domain. The method is based on the Newton-Raphson algorithm and is capable of finding roots even when the matrix is defective. Computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available.     

Construction of Real Band Anti-Symmetric Matrices from Spectral Data

In this paper,we describe how to construct a real anti-symmetric(2p-1)-band matrix with prescribed eigenvalues in its ρ leading principal submatrices.This is done in two steps.First,an anti-symmetric matrix B is constructed with the specified spectral data but not necessary a band matrix.Then B is transformed by Householder transformations to a (2ρ-1)-band matrix with the prescribed eigenvalues.An algorithm is presented.Numerical results are presented to demonstrate that the proposed method is effective.     

Reconstruction of an Impedance in Two-dimensions from Spectral Data

The two-dimensional spectral inverse problem involves the reconstruction of an unknown coefficient in an elliptic partial differential equation from spectral data, such as eigenvalues. Projection of the boundary value problem and the unknown coefficient onto appropriate vector spaces leads to a matrix inverse problem. Unique solutions of this matrix inverse problem exist provided that the eigenvalue data is close to the eigenvalues associated with the analogous constant coefficient boundary value problem. We discuss here the application of such a technique to the reconstruction of an impedance p in the boundary value problem $$ \eqalign{ -\nabla (\,p \nabla u) = \lambda p u \hbox {\quad in R} \cr u = 0 \hbox {\quad on R}}$$ where R is a rectangular domain. The matrix inverse problem, although nonstandard, is solved by a fixed-point iterative method and an impedance function p * is constructed which has the same m lowest eigenvalues as the unknown p . Numerical evidence of the success of the method will be presented.     

带比例关系的实带状矩阵特征值反问题

研究了通过矩阵A的顺序主子矩阵A_((k))=(aij)_(i,j=1)^(n-k+1)的特征值{λ_i(n-k+1)的特征值{λ_i^((k)))}_(i=1)((k)))}_(i=1)^(n-k+1)k=1,2,…,r+1来构造一个带比例关系的实带状矩阵的特征值反问题.对当特征值{λ_i(n-k+1)k=1,2,…,r+1来构造一个带比例关系的实带状矩阵的特征值反问题.对当特征值{λ_i^((k))}_(i=1)((k))}_(i=1)^(n-k+1)中有多重特征值出现时,应当如何来构造这类矩阵进行了讨论,并给出了问题的具体算法及数值例子.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

A note on quaternion matrices and split quaternion matrix pencils

In this paper, localization theorems for left and right eigenvalues of a quaternion matrix are presented. Some differences between quaternion matrices and split quaternion matrices are summarized. A counter example for Gerschgorin theorems for left and right eigenvalues of a split quaternion matrix is given. Finally, a method for finding right eigenvalues of a split quaternion matrix pencil is presented.     

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.     

Computing the matrix sign and absolute value functions

We present two algorithms for the computation of the matrix sign and absolute value functions. Both algorithms avoid a complete diagonalisation of the matrix, but they however require some informations regarding the eigenvalues location. The first algorithm consists in a sequence of polynomial iterations based on appropriate estimates of the eigenvalues, and converging to the matrix sign if all the eigenvalues are real. Convergence is obtained within a finite number of steps when the eigenvalues are exactly known. Nevertheless, we present a second approach for the computation of the matrix sign and absolute value functions, when the eigenvalues are exactly known. This approach is based on the resolution of an interpolation problem, can handle the case of complex eigenvalues and appears to be faster than the iterative approach. To cite this article: M. Ndjinga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

求非亏损矩阵特征值的一种数值计算方法

借助相似变换将非亏损矩阵转为Hessenberg矩阵,通过获得确定Hessenberg矩阵特征多项式系数的方法,利用特征值与特征多项式系数间的关系,给出求非亏损矩阵特征值的一种数值算法。     

矩阵特征值的一类新的包含域

用盖尔圆盘定理来估计矩阵的特征值是一个经典的方法,这种方法仅利用矩阵的元素来确定特征值的分布区域.本文利用相似矩阵有相同的特征值这一理论,得到了矩阵特征值的一类新的包含域,它们与盖尔圆盘等方法结合起来能提高估计的精确度.     

Integrable discrete hungry systems and their related matrix eigenvalues

Recently, some of the authors designed an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain class of band matrix. The recursion formula of the dhLV algorithm is based on the discrete hungry Lotka–Volterra (dhLV) system, which is an integrable system. One of the authors has proposed an algorithm, named the multiple dqd algorithm, for computing eigenvalues of a totally nonnegative (TN) band matrix. In this paper, by introducing a theorem on matrix eigenvalues, we first show that the eigenvalues of a TN matrix are also computable by the dhLV algorithm. We next clarify the asymptotic behavior of the discrete hungry Toda (dhToda) equation, which is also an integrable system, and show that a similarity transformation for a TN matrix is given through the dhToda equation. Then, by combining these properties of the dhToda equation, we design a new algorithm, named the dhToda algorithm, for computing eigenvalues of a TN matrix. We also describe the close relationship among the above three algorithms and give numerical examples.     

An eigenvalue problem for a symmetric Toeplitz matrix

An algorithm is developed which determines eigenvalues for a symmetric Toeplitz matrix. To this end, we substantiate the generality of eigenvalues problems for a symmetric Toeplitz matrix and for a persymmetric Hankel one. The latter is reduced to an eigenvalue problem for a persymmetric Jacobi matrix. In the even order case, the problem reduces to a Jacobi matrix with halved order.     

生成度量矩阵的新算法及拟度量矩阵的性质研究

阐述了生成度量矩阵的除法变换的新算法,这种算法使计算度量矩阵更简便,使数据的存贮空间大大减少.另外,得到了拟度量矩阵和拟判断矩阵的特征值和特征向量的关系,以及对广义成对比较矩阵和广义度量矩阵的性质进行了研究,这为不严格的成对比较矩阵的权重研究提供了较好的理论方法,为模糊矩阵的研究提供了较好的思想.     

The inverse eigenvalue problem for symmetric quasi anti-bidiagonal matrices

In this paper we construct the symmetric quasi anti-bidiagonal matrix that its eigenvalues are given, and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues. Not only elements of the tridiagonal matrix come from quasi anti-bidiagonal matrix, but also the places of elements exchange based on some conditions.     

Inverse scattering transform for the two-component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark-dark solitons

The two-component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.     

The Moore-Penrose inverse of a retrocirculant

In a recent paper Chao [2] has determined the eigenvalues of a matrix of the form A=PC where P is a permutation matrix which commutes with a certain unitary matrix and C is a circulant. Here we determine the Moore-Penrose inverse of such a “retrocirculant” and show that the nonzero eigenvalues of the Moore-Penrose inverse are the reciprocals of the nonzero eigenvalues of the retrocirculant.