scholarly journals Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede

2018 ◽  
Vol 6 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Debashish Sharma ◽  
Mausumi Sen
Keyword(s):  
Eigenvalue Problem ◽  
Recurrence Relations ◽  
Eigenvalue Problems ◽  
Inverse Eigenvalue Problem ◽  
Characteristic Polynomials ◽  
Inverse Eigenvalue Problems ◽  
Inverse Eigenvalue ◽  
Principal Submatrices ◽  
Arrow Matrix ◽  
New System

Abstract The reconstruction of a matrix having a pre-defined structure from given spectral data is known as an inverse eigenvalue problem (IEP). In this paper, we consider two IEPs involving the reconstruction of matrices whose graph is a special type of tree called a centipede. We introduce a special type of centipede called dense centipede.We study two IEPs concerning the reconstruction of matrices whose graph is a dense centipede from given partial eigen data. In order to solve these IEPs, a new system of nomenclature of dense centipedes is developed and a new scheme is adopted for labelling the vertices of a dense centipede as per this nomenclature . Using this scheme of labelling, any matrix of a dense centipede can be represented in a special form which we define as a connected arrow matrix. For such a matrix, we derive the recurrence relations among the characteristic polynomials of the leading principal submatrices and use them to solve the above problems. Some numerical results are also provided to illustrate the applicability of the solutions obtained in the paper.

scholarly journals Extremal Inverse Eigenvalue Problem for a Special Kind of Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhibing Liu ◽  
Yeying Xu ◽  
Kanmin Wang ◽  
Chengfeng Xu
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Eigenvalue ◽  
Maximal Eigenvalues ◽  
Principal Submatrices ◽  
Necessary And Sufficient ◽  
Arrow Matrix

We consider the following inverse eigenvalue problem: to construct a special kind of matrix (real symmetric doubly arrow matrix) from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. Our results are constructive and they generate algorithmic procedures to construct such matrices.

scholarly journals Inverse Eigenvalue Problems for Singular Rank One Perturbations of a Sturm-Liouville Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Xuewen Wu
Keyword(s):  
Eigenvalue Problem ◽  
Potential Function ◽  
Liouville Operator ◽  
Eigenvalue Problems ◽  
Inverse Eigenvalue Problem ◽  
Inverse Eigenvalue Problems ◽  
Rank One ◽  
Inverse Eigenvalue ◽  
Sturm Liouville

This paper is concerned with the inverse eigenvalue problem for singular rank one perturbations of a Sturm-Liouville operator. We determine uniquely the potential function from the spectra of the Sturm-Liouville operator and its rank one perturbations.

A CG-Type Method for Inverse Quadratic Eigenvalue Problems in Model Updating of Structural Dynamics

2011 ◽  
Vol 3 (1) ◽  
pp. 65-86
Author(s):  
Jiaofen Li ◽  
Xiyan Hu
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Model Updating ◽  
Inverse Eigenvalue Problem ◽  
Computationally Efficient ◽  
Type Method ◽  
Quadratic Pencil ◽  
Model Coefficient ◽  
Inverse Eigenvalue ◽  
The Difference

AbstractIn this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matricesM, DandKfor the quadratic pencilQ(λ) =λ2M+ λD+K, so thatQ(λ) has a prescribed subset of eigenvalues and eigenvectors. This method can determine the solvability of the inverse eigenvalue problem automatically. We then consider the least squares model for updating a quadratic pencilQ(λ). More precisely, we update the model coefficient matrices M, C and K so that (i) the updated model reproduces the measured data, (ii) the symmetry of the original model is preserved, and (iii) the difference between the analytical triplet (M, D, K) and the updated triplet (Mnew,Dnew,Knew) is minimized. In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.

Structured inverse eigenvalue problems

Acta Numerica ◽  
2002 ◽  
Vol 11 ◽  
pp. 1-71 ◽  
Author(s):  
Moody T. Chu ◽  
Gene H. Golub
Keyword(s):  
Eigenvalue Problem ◽  
Spectral Data ◽  
Physical System ◽  
Eigenvalue Problems ◽  
Partial Information ◽  
Normal Modes ◽  
Dynamical Behaviour ◽  
Inverse Eigenvalue Problem ◽  
Feasibility Constraints ◽  
Inverse Eigenvalue

An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behaviour. Spectral information is entailed because the dynamical behaviour is often governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system is often subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure embodied by the matrices can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well as the given spectral property. In this expository paper the emphasis is to provide an overview of the vast scope of this intriguing problem, treating some of its many applications, its mathematical properties, and a variety of numerical techniques.

On inverse eigenvalue problems for two kinds of special banded matrices

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 371-385 ◽  
Author(s):  
Wei-Ru Xu ◽  
Guo-Liang Chen
Keyword(s):  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Numerical Algorithms ◽  
Tridiagonal Matrices ◽  
Inverse Eigenvalue Problems ◽  
Banded Matrices ◽  
Inverse Eigenvalue ◽  
Maximal Eigenvalues ◽  
Principal Submatrices ◽  
Necessary And Sufficient

This paper presents two kinds of symmetric tridiagonal plus paw form (hereafter TPPF) matrices, which are the combination of tridiagonal matrices and bordered diagonal matrices. In particular, we exploit the interlacing properties of their eigenvalues. On this basis, the inverse eigenvalue problems for the two kinds of symmetric TPPF matrices are to construct these matrices from the minimal and the maximal eigenvalues of all their leading principal submatrices respectively. The necessary and sufficient conditions for the solvability of the problems are derived. Finally, numerical algorithms and some examples of the results developed here are given.

Design of Nonproportional Damped Systems via Symmetric Positive Inverse Problems

2004 ◽  
Vol 126 (2) ◽  
pp. 212-219 ◽  
Author(s):  
L. Starek ◽  
D. J. Inman
Keyword(s):  
Eigenvalue Problems ◽  
Positive Definite ◽  
Inverse Eigenvalue Problem ◽  
Eigenvalues And Eigenvectors ◽  
Inverse Eigenvalue Problems ◽  
Complex Eigenvalues ◽  
Inverse Eigenvalue ◽  
Nonproportional Damping ◽  
Damped Systems ◽  
Vibrating Systems

This paper summarizes the authors’ previous efforts on solving inverse eigenvalue problems for linear vibrating systems described by a vector differential equations with constant coefficient matrices and nonproportional damping. The inverse problem of interest here is that of determining symmetric, real, positive definite coefficient matrices assumed to represent mass normalized velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. Two previous solutions to the symmetric inverse eigenvalue problem, presented by Starek and Inman, are reviewed and then extended to the design of underdamped vibrating systems with nonproportional damping.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

scholarly journals A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Weiping Shen
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Eigenvalue Problems ◽  
Generalized Newton Method ◽  
Inexact Newton Method ◽  
Numerical Tests ◽  
Inverse Eigenvalue Problems ◽  
Inverse Eigenvalue ◽  
Generalized Newton ◽  
Special Case

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

Sign in / Sign up

Export Citation Format

Share Document