Inverse Eigenvalue Problem For Matrices Research Articles

Two inverse eigenvalue problems for matrices whose graphs are trees

Inverse eigenvalue problem refers to the problem of reconstructing a matrix of a desired structure from a prescribed eigendata. In this paper, we discuss two additive inverse eigenvalue problems for matrices whose graph is tree. In order to analyze the problems, the vertices of the given tree are labeled in a suitable way so that concrete recurrence relations between the characteristic polynomials of the leading principal submatrices of the matrix can be obtained in a simple form. The method of obtaining the entries of the required matrix is constructive and provides an algorithm for computing the same. We provide some numerical examples to illustrate the results. The computations are done using SCILAB by feeding the eigendata and the adjacency pattern of the tree as inputs.

The minimax inverse eigenvalue problem for matrices whose graph is a generalized star of depth 2

In this paper, we study an inverse eigenvalue problem, referred to as the minimax inverse eigenvalue problem, of constructing matrices whose graph is a special type of tree called a generalized star of depth 2. A scheme of labelling the vertices of such a tree is introduced in order to express the corresponding matrices in a number of special forms. We then solve the minimax inverse eigenvalue problem of constructing such matrices from given eigen data consisting of the minimal and maximal eigenvalues of each of the leading principal submatrices of the required matrices. We pose an open problem of solving the minimax inverse eigenvalue problem for matrices described by arbitrary trees.

An inverse eigenvalue problem for modified pseudo-Jacobi matrices

In this paper, we investigate an inverse eigenvalue problem for matrices that are obtained from pseudo-Jacobi matrices by only modifying the (1,r)-th and (r,1)-th entries, 3≤r≤n. Necessary and sufficient conditions under which the problem is solvable are derived. Uniqueness results are presented and an algorithm to reconstruct the matrices from the given spectral data is proposed. Illustrative examples are provided.

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.

To solving inverse eigenvalue problems for matrices

The paper considers additive and multiplicative inverse eigenvalve problems and suggests a solution method based on the connection of inverse problems with the problem of computing the regular spectrum of a multiparameter pencil of constant matrices corresponding to the problem considered. Necessary and sufficient conditions for the solvability of inverse eigenvalve problems are established. Bibliography: 3 titles.

On Some Inverse Eigenvalue Problems with Toeplitz-Related Structure

Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application.

An Approach to Solving Inverse Eigenvalue Problems for Matrices

The paper considers different formulations of inverse eigenvalue problems for matrices whose entries either polynomially or rationally depend on unknown parameters. An approach to solving inverse problems together with numerical algorithms is suggested. The solution of inverse problems is reduced to the problem of finding the so-called discrete solutions of nonlinear algebraic systems. The corresponding systems are constructed using the method of traces, and their discrete roots are found by applying the algorithms for solving nonlinear algebraic systems in several variables previously suggested by the author. Bibliography: 30 titles.

A Note on the Symmetric Recursive Inverse Eigenvalue Problem

In [M. Arav et al., SIAM J. Matrix Anal. Appl., 22 (2000), pp. 392--412] the recursive inverse eigenvalue problem for matrices was introduced. In this paper we examine an open problem on the existence of symmetric positive semidefinite solutions that was posed there. We first give several counterexamples for the general case and then characterize under which further assumptions the conjecture is valid.

The Recursive Inverse Eigenvalue Problem

The recursive inverse eigenvalue problem for matrices is studied, where for each leading principal submatrix an eigenvalue and associated left and right eigenvectors are assigned. Existence and uniqueness results as well as explicit formulas are proven, and applications to nonnegative matrices, Z-matrices, M-matrices, symmetric matrices, Stieltjes matrices, and inverse M-matrices are considered.

New conditions for the additive inverse eigenvalue problem for matrices

This paper deals with the following inverse eigenvalue problem: Given an n by n real symmetric matrix A and a set of real numbers { λ i } 1 n , find a diagonal matrix D such that A + D has eigenvalues λ i . For the solvability of this problem a number of necessary conditions and sufficient conditions are known. In this work, new necessary conditions are derived, while some sufficient conditions are optimized. In particular, one sufficient condition due to Morel is obtained by optimizing a sufficient condition discovered by Laborde.

On the inverse eigenvalue problem for matrices (atomic corrections)

The inverse eigenvalue problem for matrices is studied with the objective of obtaining an efficient method for correcting energy levels in atomic systems, though the results are applicable to any eigenvalue problem. The approach is a development of earlier work by S. Friedland (1977). The diagonal elements of a real symmetric matrix with given off-diagonal elements are adjusted to yield a given spectrum. The authors discuss cases where there are real solutions and no real solutions, with particular emphasis on the latter. Problems of slow convergence arise. They demonstrate the cause of this slow convergence, give a geometrical interpretation of the problem and show how it can be avoided. Also the matrices encountered arise from complex calculations and are subject to error. They develop an error analysis that permits us, among other things, to judge whether corrections in any particular case are justified in view of anticipated errors in the given computed off-diagonal matrix elements. Finally, the method is demonstrated in an application to certain sets of levels in 12 times ionized (neon-like) titanium.

Inverse multiparameter eigenvalue problems for matrices II

This is a sequel to our previous paper [4] where we initiated a study of inverse eigenvalue problems for matrices in the multiparameter setting. The one parameter version of the problem under consideration asks for conditions on a given n × n symmetric matrix A and on n given real numbers s1≦s2≦…≦sn under which a diagonal matrix V can be found so that A + V has sl,…,sn as its eigenvalues. Our motivation for this problem and our method of attack on it in [4]p comes chiefly from the work of Hadeler [5] in which sufficient conditions were given for existence of the desired diagonal V. Hadeler's approach in [5] relied heavily on the Brouwer fixed point theorem and this was also our main tool in [4]. Subsequently, using properties of topological degree, Hadeler [6] gave somewhat different conditions for the existence of the diagonal V. It is our desire here to follow this lead and to use degree theory to give some results extending those in [6] to the multiparameter case.

Inductor-capacitor one-ports and inverse eigenvalue problems

The newly discovered canonicity theorem for LC one-ports, which states conditions for such one-ports to be able to realize arbitrary, positive-real-odd impedance functions with the minimum number of components, is shown to provide conditions for the existence of a solution to a special, inverse-eigenvalue problem for matrices. Certain recent results in matrix theory are proved to be particular solutions of this inverse-eigenvalue problem, and these solutions are then shown to provide new and independent canonicity proofs for the Foster and Cauer forms. The results derived herein may prove useful in developing numerical design methods for the general class of canonic LC one-ports.

Inverse eigenvalue problems

The classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied. Using general results on the solvability of polynomial systems it is shown that in the complex case these problems are always solvable by a finite number of solutions. In case of real symmetric matrices the inverse problems are reformulated to have a real solution. An algorithm is given to obtain this solution.