Eigenvalue Problem For Matrices Research Articles

A numerical projection technique for large-scale eigenvalue problems

We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity is constructed by projecting out high energy degrees of freedom and in turn solving the resulting model by some standard eigenvalue solver.Here we introduce a generalization of this idea, where both steps are performed numerically and which in contrast to the standard projection technique converges in principle to the exact eigenvalues. This approach is not just applicable to eigenvalue problems encountered in many-body systems but also in other areas of research that result in large-scale eigenvalue problems for matrices which have, roughly speaking, mostly a pronounced dominant diagonal part. We will present detailed studies of the approach guided by two many-body models.

Convergence Analysis of Gradient Iterations for the Symmetric Eigenvalue Problem

Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poorest convergence of the eigenvalue approximations is attained in a three-dimensional invariant subspace; explicit convergence estimates are then derived by means of a minidimensional analysis.

Perturbation of Partitioned Hermitian Definite Generalized Eigenvalue Problems

This paper is concerned with the Hermitian definite generalized eigenvalue problem for block diagonal matrices and . Both and are Hermitian, and is positive definite. Bounds on how its eigenvalues vary when and are perturbed by Hermitian matrices are established. These bounds are generally of linear order with respect to the perturbations in the diagonal blocks and of quadratic order with respect to the perturbations in the off-diagonal blocks. The results for the case of no perturbations in the diagonal blocks can be used to bound the changes of eigenvalues of a Hermitian definite generalized eigenvalue problem after its off-diagonal blocks are dropped, a situation that occurs frequently in eigenvalue computations. The presented results extend those of Li and Li [Linear Algebra Appl., 395 (2005), pp. 183–190]. It was noted by Stewart and Sun [Matrix Perturbation Theory, Academic Press, Boston, 1990] that different copies of a multiple eigenvalue may exhibit quite different sensitivities towards perturbations. We establish bounds to reflect that feature, too. We also derive quadratic eigenvalue bounds for diagonalizable non-Hermitian pencils subject to off-diagonal perturbations.

Many fields interaction: Beam splitters and waveguide arrays

AbstractWe study the interaction of many fields. We obtain an effective Hamiltonian for this system by using a method recently introduced that produces a small rotation to the Hamiltonian that allows to neglect some terms in the rotated Hamiltonian. We show that coherent states remain coherent under the action of a quadratic Hamiltonian and by solving the eigenvalue and eigenvector problem for tridiagonal matrices we also show that a system of n interacting harmonic oscillators, initially in coherent states, remain coherent during the interaction.

An overview on the eigenvalue computation for matrices

This paper sketches the research developments in the area of computational methods for solving the eigenvalue problems and how the methods developed relate to each other as they evolve over centuries. This is an attempt to write a complete overview on the research on computational aspects of eigenvalue problem, emphasize the history of methods that still play a role and some of those that no longer are considered to be on the main track but are somehow related to the present techniques in some smaller steps. This contribution brings out the state-of-the-art of the algorithms for solving large-scale eigenvalue problems for both symmetric and nonsymmetric matrices separately, thereby clearly drawing a comparison between the differences in the algorithms in practical use for the two. Some of the methods or modifications to the earlier methods that have been notable at the turn of the 21st century will also be covered through this paper under Modem Approaches. Also as the standard eigenvalue problem has become better understood, in a numerical sense, progress has been made in the generalized eigenvalue problem and this will briefly cover developments in solving these problems too.

Inverse eigenvalue problems for bisymmetric matrices under a central principal submatrix constraint

This article considers an inverse eigenvalue problem for bisymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

A uniform object-oriented solution to the eigenvalue problem for real symmetric and Hermitian matrices

We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them. Program summary Program title: SHMatrix Catalogue identifier: AEHZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2616 No. of bytes in distributed program, including test data, etc.: 127 312 Distribution format: tar.gz Programming language: Standard ANSI C++. Computer: PCs and workstations. Operating system: Linux, Windows. Classification: 4.8. Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide and conquer, can be used to factorize the matrices in order to apply a parallel computing approach. However, it is still interesting to have a core procedure able to tackle the computation of eigenvalues and eigenvectors once the matrix has been factorized to pieces of enough small size. Several available software packages, such as LAPACK, tackled this problem under the traditional imperative programming paradigm. In order to ease the modelling of complex quantum mechanical models it could be interesting to apply an object-oriented approach to the treatment of the eigenproblem. This approach offers the advantage of a single, uniform treatment for the real symmetric and Hermitian cases. Solution method: To reach the above goals, we have developed a system of classes: SHMatrix. SHMatrix is composed by an abstract base class and two descendant classes, one for real symmetric matrices and the other for the Hermitian case. The object-oriented characteristics of inheritance and polymorphism allows handling both cases using a single reference of the base class. The basic computing strategy applied in SHMatrix allows computing subsets of eigenvalues and (optionally) eigenvectors. The tests performed show that SHMatrix is competitive, and more efficient for large matrices, than the equivalent routines of the LAPACK package. Running time: The examples included in the distribution take only a couple of seconds to run.

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.

The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles

In 1979, Ferguson characterized the periodic Jacobi matrices with given eigenvalues and showed how to use the Lanzcos Algorithm to construct each such matrix. This article provides general characterizations and constructions for the complex analogue of periodic Jacobi matrices. As a consequence of the main procedure, we prove that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2.

An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle-Handelman conjecture

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A A is an ( n + 1 ) × ( n + 1 ) (n+1)\times (n+1) nonnegative matrix whose nonzero eigenvalues are: λ 0 ≥ | λ i | \lambda _0 \geq |\lambda _i| , i = 1 , … , r i=1,\ldots ,r , r ≤ n r \leq \ n , then for all x ≥ λ 0 x \geq \lambda _0 , ( ∗ ) ∏ i = 0 r ( x − λ i ) ≤ x r + 1 − λ 0 r + 1 . \begin{equation} \prod _{i=0}^{r} (x-\lambda _i) \leq x^{r+1}-\lambda _0^{r+1}.\tag *{$(\ast )$} \end{equation} To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2 ( r + 1 ) ≥ ( n + 1 ) 2(r+1)\geq (n+1) , while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 n\leq 4 and when the spectrum of A A is real. They also showed that the conjecture is asymptotically true with the dimension. Here we prove a slightly stronger inequality than in ( ∗ ) (\ast ) , from which it follows that the Boyle–Handelman conjecture is true. Actually, we do not start from the assumption that the λ i \lambda _i ’s are eigenvalues of a nonnegative matrix, but that λ 1 , … , λ r + 1 \lambda _1,\ldots , \lambda _{r+1} satisfy λ 0 ≥ | λ i | \lambda _0\geq |\lambda _i| , i = 1 , … , r i=1,\ldots , r , and the trace conditions: ( ∗ ∗ ) ∑ i = 0 r λ i k ≥ 0 , for all k ≥ 1. \begin{equation} \sum _{i=0}^{r} \lambda _i^k \geq 0, \ \mbox {for all} k \geq 1.\tag *{$(\ast \ast )$} \end{equation} A strong form of the Boyle–Handelman conjecture, conjectured in 2002 by the present authors, says that ( ∗ * ) continues to hold if the trace inequalities in ( ∗ ∗ ** ) hold only for k = 1 , … , r k=1,\ldots ,r . We further improve here on earlier results of the authors concerning this stronger form of the Boyle–Handelman conjecture.

Orthogonal polynomials from Hermitian matrices

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.

An Experimental Approach to Mathematical Modeling in Biology*

The simplest age-structured population models update a population vector via multiplication by a matrix. These linear models offer an opportunity to introduce mathematical modeling to students of limited mathematical sophistication and background. We begin with a detailed discussion of mathematical modeling, particularly in a biological context. We then describe Bugbox-population, a virtual insect laboratory that allows students to make observations and collect quantitative data easily, thereby learning mathematical modeling in the context of its use in scientific research. Creating a mathematical model for boxbugs involves the same intellectual work as creating a mathematical model for real insects, but without the difficulties involved in collecting real biological data. The analysis of the Bugbox-population data leads to the development of the eigenvalue problem for population projection matrices.

The inverse eigenvalue problem of reflexive matrices with a submatrix constraint and its approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenvalue problem defined as follows: given a generalized reflection matrix P∈Rn×n, a set of complex n-vectors {xi}i=1m, a set of complex numbers {λi}i=1m, and an s-by-s real matrix C0, find an n-by-n real reflexive matrix C such that the s-by-s leading principal submatrix of C is C0, and {xi}i=1m and {λi}i=1m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix \(\tilde{C}\) , find a matrix C which is the solution to the constrained inverse problem such that the distance between C and \(\tilde{C}\) is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. An illustrative experiment is also presented.

Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving‐point immittance functions of transformerless, common‐ground, two‐element‐kind RLC networks and in generation of their equivalent realizations.

The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs

The nonnegative inverse eigenvalue problem (NIEP) is: given a family of complex numbers σ = { λ 1 , … , λ n } , find necessary and sufficient conditions for the existence of a nonnegative matrix A of order n with spectrum σ . Loewy and London [R. Loewy, D. London, A note on the inverse eigenvalue problems for nonnegative matrices, Linear and Multilinear Algebra 6 (1978) 83–90] resolved it for n = 3 , and for n = 4 when the spectrum is real. In our way of handling the NIEP, we focus our attention on the coefficients of the characteristic polynomial of A. Thus, the NIEP that we consider is: “ given k 1 , k 2 , … , k n real numbers, find necessary and sufficient conditions for the existence of a nonnegative matrix A of order n with characteristic polynomial x n + k 1 x n - 1 + k 2 x n - 2 + ⋯ + k n ”. The coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix A. We introduce a special type of digraph structure, that we shall call EBL, in which this relation is specially simple. We give some results that show the interest of EBL structures. We completely solve the NIEP from the coefficients of the characteristic polynomial for n = 4 . We also solve a special case of the NIEP for n ⩽ 2 p + 1 with k 1 = ⋯ = k p - 1 = 0 and p ⩾ 2 .

Multiple-scattering and electron-uracil collisions at low energies

Links between two well known methods: methods of zero-range and non-overlapped (muffin-tin) potentials are discussed. Some difficulties of the method of zero-range potentials and its possible elimination are discussed. We argue that such advanced method of ZRP potential can be applied to realistic electron-molecular processes. The method reduces electron-molecule scattering to generalized eigenvalue problem for hermitian matrices and admit fast numerical scheme. A noteworthy feature of the method is direct possibility to calculate the wave functions (partial waves). The theory is applied to electron-uracil scattering. Partial phases and cross-sections at low energies are evaluated and plotted.

To solving inverse eigenvalue problems for parametric matrices

Some statements of inverse eigenvalue problems for one-parameter and multiparameter regular polynomial matrices with linear and nonlinear dependences on spectral parameters are considered. Methods for solving inverse eigenvalue problems based on rank factorization, exhaustion, and reduction to nonlinear equations are proposed. Bibliography: 12 titles.

An efficient algorithm for generalized discriminant analysis using incomplete Cholesky decomposition

Generalized discriminant analysis (GDA) has provided an extremely powerful approach to extracting nonlinear features via kernel trick. And it has been suggested for a number of applications, such as classification problem. Whereas the GDA could be solved by the utilization of Mercer kernels, a drawback of the standard GDA is that it may suffer from computational problem for large scale data set. Besides, there is still attendant problem of numerical accuracy when computing the eigenvalue problem of large matrices. Also, the GDA would occupy large memory (to store the kernel matrix). To overcome these deficiencies, we use Gram–Schmidt orthonormalization and incomplete Cholesky decomposition to find a basis for the entire training samples, and then formulate GDA as another eigenvalue problem of matrix whose size is much smaller than that of the kernel matrix by using the basis, while still working out the optimal discriminant vectors from all training samples. The theoretical analysis and experimental results on both artificial and real data set have shown the superiority of the proposed method for performing GDA in terms of computational efficiency and even the recognition accuracy, especially when the training samples size is large.

Class-Incremental Generalized Discriminant Analysis

Generalized discriminant analysis (GDA) is the nonlinear extension of the classical linear discriminant analysis (LDA) via the kernel trick. Mathematically, GDA aims to solve a generalized eigenequation problem, which is always implemented by the use of singular value decomposition (SVD) in the previously proposed GDA algorithms. A major drawback of SVD, however, is the difficulty of designing an incremental solution for the eigenvalue problem. Moreover, there are still numerical problems of computing the eigenvalue problem of large matrices. In this article, we propose another algorithm for solving GDA as for the case of small sample size problem, which applies QR decomposition rather than SVD. A major contribution of the proposed algorithm is that it can incrementally update the discriminant vectors when new classes are inserted into the training set. The other major contribution of this article is the presentation of the modified kernel Gram-Schmidt (MKGS) orthogonalization algorithm for implementing the QR decomposition in the feature space, which is more numerically stable than the kernel Gram-Schmidt (KGS) algorithm. We conduct experiments on both simulated and real data to demonstrate the better performance of the proposed methods.

Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices

Presented here are two related numerical methods, one for the inverse eigenvalue problem for nonnegative or stochastic matrices and another for the inverse eigenvalue problem for symmetric nonnegative matrices. The methods are iterative in nature and utilize alternating projection ideas. For the algorithm for the symmetric problem, the main computational component of each iteration is an eigenvalue-eigenvector decomposition, while for the other algorithm, it is a Schur matrix decomposition. Convergence properties of the algorithms are investigated and numerical results are also presented. While the paper deals with two specific types of inverse eigenvalue problems, the ideas presented here should be applicable to many other inverse eigenvalue problems, including those involving nonsymmetric matrices.