Inverse Eigenvalue Problem Research Articles

Inverse eigenvalue problem of Jacobi matrix with mixed data

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from its eigenvalues, its leading principal submatrix and part of the eigenvalues of its submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.

Inverse Eigenvalue Problem for a Class of Upper Triangular Matrices with Linear Relation

This paper studies on the eigenvalue[1-5] of the a class of upper triangular matrix with linear relation. It discusses the feature of existence and uniqueness of matrix via two given two characteristic pairs(λ,χ),(μ,γ) . Solutions and expressions are provided under satisfied conditions. The possibilities are exanimated by numerical example.

A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues

In this paper, we grasp an inverse eigenvalue problem which constructs a tridiagonal matrix with specified multiple eigenvalues, from the viewpoint of the quotient difference (qd) recursion formula. We also prove that the characteristic and the minimal polynomials of a constructed tridiagonal matrix are equal to each other. As an application of the qd formula, we present a procedure for getting a tridiagonal matrix with specified multiple eigenvalues. Examples are given through providing with four tridiagonal matrices with specified multiple eigenvalues.

A structured method for the design-for-frequency of a brace-soundboard system using a scalloped brace

Design-for-frequency of mechanical systems has long been a practice of iterative procedures in order to construct systems having desired natural frequencies. Especially problematic is achieving acoustic consistency in systems using natural materials such as wood. Inverse eigenvalue problem theory seeks to rectify these shortfalls by creating system matrices of the mechanical systems directly from the desired natural frequencies. In this paper, the Cayley–Hamilton and determinant methods for solving inverse eigenvalue problems are applied to the problem of the scalloped braced plate. Both methods are shown to be effective tools in calculating the dimensions of the brace necessary for achieving a desired fundamental natural frequency and one of its higher partials. These methods use the physical parameters and mechanical properties of the material in order to frame the discrete problem in contrast to standard approaches that specify the structure of the matrix itself. They also demonstrate the ability to find multiple solutions to the same problem. The determinant method is found to be computationally more efficient for partially described inverse problems due to the reduced number of equations and parameters that need to be solved. The two methods show great promise for techniques which could lead to the design of complex mechanical systems, including musical instrument soundboards, directly from knowledge of the desired natural frequencies.

On Sufficient Conditions for the RNIEP and their Margins of Realizability

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) is that of characterizing all possible real spectra of nonnegative matrices. In this work we list some inclusion relations between several sufficient conditions and we study the negativity and the realizability margin of a spectrum with respect to these conditions.

Normal nonnegative realization of spectra

The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353–364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu’s conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.

Application of the alternating direction method for an inverse monic quadratic eigenvalue problem

This paper shows that the alternating direction method can be used to solve the inverse monic quadratic eigenvalue problem (IMQEP) with the semidefiniteness and sparsity requirements. The results of numerical examples show that the proposed method works well.

On the generalized inverse eigenvalue problem of constructing symmetric pentadiagonal matrices from three mixed eigendata

In this paper, we consider a generalized inverse eigenvalue problem of the form , where and are both pentadiagonal matrices. We propose an algorithm for reconstructing a pentadiagonal matrix . Let be leading principal submatrix of . Given , a pentadiagonal matrix , distinct real numbers , , and real vectors , , , where , we construct pentadiagonal matrix and three vectors , and such that is leading principal submatrix of and , and are eigenpairs of , where , and . Sufficient conditions for solvability of the problem are given.

An analytic mapping of oligomer potential energy surfaces to an effective Frenkel model.

While the use of Frenkel-type models for semiconducting polymer assemblies and related molecular aggregates is well established, the direct parametrization of such models based on electronic structure data is attempted less frequently. In this work, we develop a systematic mapping procedure which is adapted to J-type and H-type homo-aggregate systems. The procedure is based upon the analytic solution of an inverse eigenvalue problem for an effective Frenkel Hamiltonian with nearest-neighbor couplings. Vibronic interactions are included for both site-local and site-correlated modes. For illustration, an application is presented to the excited-state ab initio potential energy surfaces (PESs) of an oligothiophene octamer. The procedure performs a pointwise mapping of the PESs of oligomers of arbitrary chain length n, provided that the electronic ground state and any two of the n lowest adiabatic states of the excitonic manifold of interest are known. These three states are reproduced exactly by the procedure while the remaining n - 2 states of the excitonic manifold can be predicted. Explicit conditions are derived permitting to verify whether a given data set is compatible with the effective Frenkel model under study.

A structured approach to design-for-frequency problems using the Cayley-Hamilton theorem.

An inverse eigenvalue problem approach to system design is considered. The Cayley-Hamilton theorem is developed for the general case involving the generalized eigenvalue vibration problem. Since many solutions exist for a desired frequency spectrum, a discussion of the required design information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue design problem using the generalized Cayley-Hamilton theorem is proposed. A method for solving partially described systems is also specified. The Cayley-Hamilton theorem algorithm is shown to be a good design tool for solving inverse eigenvalue problems of mechanical and structural systems.

Random inverse spectral problems and closed random exponential systems

We study the inverse eigenvalue problem for the half-line random Schrödinger operators. Generalizing results from Miklòs Horvàth, we obtain optimal and almost optimal conditions for a set of eigenvalues to determine the random Schrödinger operator almost precisely. These conditions are simple closed properties of the random exponential system corresponding to the known eigenvalues.

Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient

In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.

Graphs that allow all the eigenvalue multiplicities to be even

Let G be an undirected graph on n vertices and let S(G) be the set of all n×n real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The Inverse Eigenvalue Problem for a graph G is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in S(G). This question is, in general, hard to answer and several variations were studied, most notably the minimum rank problem. In this paper we introduce the problem of determining for which graphs G there exists a matrix in S(G) whose characteristic polynomial is a square, i.e. the multiplicities of all its eigenvalues are even. We solve this question for several families of graphs.

Optimization problems on the rank of the solution to left and right inverse eigenvalue problem

A complex matrix $P$ is called Hermitian and $\{k+1\}$-potent if $P^{k+1}=P=P^*$ for some integer $k\geq 1$. Let $P$ and $Q$ be $n\times n$ Hermitian and $\{k+1\}$-potent matrices, we say that complex matrix $A$ is $\{P,Q,k+1\}$-reflexive (anti-reflexive) if $PAQ=A$ ($PAQ=-A$). In this paper, the solvability conditions and the general solutions of the left and right inverse eigenvalue problem for $\{P,Q,k+1\}$-reflexive and anti-reflexive matrices are derived, and the minimal and maximal rank solutions are given. Moreover, the associated optimal approximation problem is also considered. Finally, numerical example is given to illustrate the main results.

On the inverse eigenvalue problem for T-alternating and T-palindromic matrix polynomials

The inverse eigenvalue problem for T-alternating matrix polynomials over arbitrary algebraically closed fields of characteristic different from two is considered. The main result shows that the necessary conditions obtained in [9] for a matrix polynomial to be the Smith form of a T-alternating matrix polynomial are under mild conditions also sufficient to be the Smith form of a T-alternating matrix polynomial with invertible leading coefficient which is additionally in anti-triangular form. In particular, this result implies that any T-alternating matrix polynomial with invertible leading coefficient is equivalent to a T-alternating matrix polynomial in anti-triangular form that has the same finite and infinite elementary divisors as the original matrix polynomial. Finally, the inverse eigenvalue problem for T-palindromic matrix polynomials is considered excluding the case that both +1 and −1 are eigenvalues.

Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

In inverse eigenvalue problems one tries to reconstruct a matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved.First, given the eigenvalues and the first components of the associated eigenvectors (called the weight vector) an extended Hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints. The extended Hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues. This inverse problem closely links to orthogonal rational functions: the extended Hessenberg matrix contains the recurrence coefficients given the nodes (eigenvalues), poles (poles of the extended Hessenberg matrix), and a weight vector (first eigenvector components) determining the discrete inner product. Moreover, it is also sort of the inverse of the (rational) Arnoldi algorithm: instead of using the (rational) Arnoldi method to compute a Krylov basis to approximate the spectrum, we will reconstruct the orthogonal Krylov basis given the spectral info.In the second inverse eigenvalue problem, we do the same, but refrain from unitarity. As a result we execute possibly non-unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a (non)-symmetric extended tridiagonal matrix. The algorithm will be less stable, but it will be faster, as the extended tridiagonal matrix admits a low cost factorization of O(n) (n equals the number of eigenvalues), whereas the extended Hessenberg matrix does not. Again there is a close link with orthogonal function theory, the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions. Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm.

Structured least-squares problems and inverse eigenvalue problems for [formula omitted]-reflexive matrices

A new meaningful structured matrix—(P,Q)-reflexive matrix is defined. Without the common assumption that P or Q is unitary, a general solution is derived for its structured least-squares problem. As a necessary and sufficient condition being presented for the solvability of its structured inverse eigenvalue problem, structured constrains are firstly given to guarantee the existence of the solution. The optimal approximation problem is also considered under spectral constrains.

General Solutions for a Class of Inverse Quadratic Eigenvalue Problems

AbstractBased on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where n × n real symmetric matrices M, C and K are constructed so that the quadratic pencil Q(λ) = λ2M + λC + K yields good approximations for the given k eigenpairs. We discuss the case where M is positive definite for 1 ≤ k ≤ n, and a general solution to this problem for n + 1 ≤ k ≤ 2n. The efficiency of our methods is illustrated by some numerical experiments.

Generalized inverse eigenvalue problem for matrices whose graph is a path

In this paper, we analyse a special generalized inverse eigenvalue problem Anx=λBnx for the pair (An,Bn) of matrices each of whose graph is a path on n vertices, by investigating the leading principal minors of the matrix An−λBn. From the given data consisting of Bn, a matrix Ak (k<n) whose graph is a path on k vertices, two column vectors X2,Y2∈Rn−k and distinct real numbers λ and μ we construct An and two column vectors X1,Y1∈Rk such that Ak is the leading principal sub-matrix of An and (λ,X), (μ,Y) are the eigenpairs of (An,Bn), where X=(X1T,X2T)T and Y=(Y1T,Y2T)T. Further, numerical examples are also given to demonstrate the applicability of the results developed here.

Quadratic inverse eigenvalue problem for damped gyroscopic systems

In this paper, we discuss the quadratic inverse eigenvalue problem for damped gyroscopic systems, that is, we are to construct a damped gyroscopic second order system having given 2 n eigenvalues a...