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.     

The Cayley Method and the Inverse Eigenvalue Problem for Toeplitz Matrices

Despite the fact that symmetric Toeplitz matrices can have arbitrary eigenvalues, the numerical construction of such a matrix having prescribed eigenvalues remains to be a challenge. A two-step method using the continuation idea is proposed in this paper. The first step constructs a centro-symmetric Jacobi matrix with the prescribed eigenvalues in finitely many steps. The second step uses the Cayley transform to integrate flows in the linear subspace of skew-symmetric and centro-symmetric matrices. No special geometric integrators are needed. The convergence analysis is illustrated for the case of n = 3. Numerical examples are presented.     

基于样条的高阶矩阵微分方程近似解

矩阵微分方程经常出现在许多物理模型和工程技术模型中.利用矩阵样条构造形如{y(p)(x)=Ap-1(x)y(p-1)(x)+Ap-2(x)y(p-2)(x)+…+A1(x)y(1)(x)+A0(x)y(x)+B0(x),y(a)=ya,…,y(p-1)(a)=y(p-1)a,x∈[a,b];Ai(x),B0(x)∈C4[a,b],0≤i≤p-烅烄烆1的高阶矩阵线性微分方程初值问题的数值解.给出实现算法和数值解的近似误差估计以及数值实例.先将高阶矩阵微分方程转化为一阶矩阵微分方程,然后利用三次矩阵样条求出一阶矩阵线性微分方程的数值解,从而解决高阶微分方程问题.     

矩阵特征值问题的Bendixson定理的改进

任一n×n矩阵A可分解为A=B C,其中B=1/2(A A~H),C=1/2(A-A~H)。Bendixson定理的主要内容是:λ_j(A)(j=1,2,…,n)落在矩形区域F上,而构成F的四个边的直线分别为x=max(λ_j(B)),x=min(λ_j(B)),y=max(-iλ_j(C)),y=min(-iλ_j(C))。本文给出用B,C的特征值和矩阵A的正规性偏离度对A的特征值的进一步估计。     

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.     

矩阵方程（A^TXA，B^TXB=（C，D）的反对称解及其最佳逼近

本利用矩阵对的标准相关分解，得到了矩阵方程(A^TXB，B^TXB)=(C，D)反对称解存在的充分必要条件及通解表达式，同时给出了解关于已知矩阵的最佳逼近．     

The inverse eigenvalue problem for a weighted helmholtz equation

Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ^* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.     

Trace class multipliers and spectral variation of normal matrices

In this article we show how to estimate the trace multiplier norm of a rank 2 matrix. As an application, an alternative proof of a theorem of Holbrook et al. (Maximal spectral distance, Linear Algebra Appl., 249 (1996) 197–205) on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

On an inverse eigenproblem for Jacobi matrices

Recently Xu [13] proposed a new algorithm for computing a Jacobi matrix of order 2n with a given n×n leading principal submatrix and with 2n prescribed eigenvalues that satisfy certain conditions. We compare this algorithm to a scheme proposed by Boley and Golub [2], and discuss a generalization that allows the conditions on the prescribed eigenvalues to be relaxed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

由谱数据构造实双反对称矩阵

讨论实完全反对称矩阵的一个特秆值反问题．研究了实完全反对称矩阵的一些特征性质，构造一个实反对称矩阵使其各阶顺序主子矩阵具有指定的特征值．证明了：给定满足一定分隔条件的两组数，存在一个实完全反对称矩阵，使其各阶中心主子矩阵具有相应的特征值．     

Diagonal elements and eigenvalues of a real symmetric matrix

A simple and constructive proof is given for the existence of a real symmetric matrix with prescribed diagonal elements and eigenvalues. Numerically implementable algorithms for constructing such a matrix are discussed.     

Extracting partial canonical structure for large scale eigenvalue problems

We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted on problems with known canonical structure and varying ill-conditioning are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

非奇异阵特征极值和条件数的近似计算

本文从共轭梯度法的公式推导出对称正定阵A与三对角阵B的相似关系,B的元素由共轭梯度法的迭代参数确定.因此,对称正定阵的条件数计算可以化成三对角阵条件数的计算,并且可以在共轭梯度法的计算中顺带完成.它只需增加O(s)次的计算量,s为迭代次数.这与共轭梯度法的计算量相比是可以忽略的.当A为非对称正定阵时,只要A非奇异,即可用共轭梯度法计算ATA的特征极值和条件数,从而得出A的条件数.对不同算例的计算表明,这是一种快速有效的简便方法.     

矩阵特征值、特征向量的确定

首先对由 A的特征值、特征向量求 A- 1 ,AT,A* ( A的伴随矩阵 )、P- 1 AP以及 A的多项式φ( A)的特征值和特征向量的结论作了个归纳 ;对相反的情形 ,我们给出了部分已有的结果 ,并通过四道例题着重讨论了如何由 φ( A)的特征值来求 A的特征值 .     

矩阵方程AXB=C的反对称解和极小范数反对称解

建立了求矩阵方程AXB=C反对称解的迭代方法.使用该方法不仅能够判断反对称解的存在性,而且在有反对称解时,能够在有限步迭代计算之后得到反对称解.选取特殊的初始矩阵,可求得极小范数反对称解.     

Solution of the partial eigenvalue problem for a regular matrix pencilD(λ) =A − λB

One presents the ALGOL procedures which implement the algorithm for the determination of the group of smallest (greatest) eigenvalues and their corresponding eigenvectors for a matrix pencil where A and B are real square matrices of simple structure. From the initial pencil one constructs a matrix C, whose eigenvalues are taken as the initial approximations to the eigenvalues from the group of the smallest (greatest) eigenvalues of the pencil. The refinement of the eigen-values is performed on the basis of the theory of perturbations. Then one determines the eigen-vectors and one computes the infinite norm of the residual. One gives ALGOL programs and test examples.     

Nonnegative matrices with prescribed spectrum and elementary divisors

In this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors, which drastically improve and contain some of the previous known conditions. We also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide if a given list is realizable with prescribed elementary divisors.