Construction of nonnegative matrices and the inverse eigenvalue problem

This article presents a technique for combining two matrices, an n?×?n matrix M and an m?×?m matrix B, with known spectra to create an (n?+?m???p)?×?(n?+?m???p) matrix N whose spectrum consists of the spectrum of the matrix M and m???p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n?+?1 complex numbers (λ₁,λ₂,λ₃,σ) from a given realizable list of n complex numbers (c ₁,c ₂,σ), where c ₁ is the Perron eigenvalue, c ₂ is a real number and σ is a list of n???2 complex numbers.     

The constrained inverse eigenvalue problem and its approximation for normal matrices

In this paper, the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered. The solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented, and the expression of the solution for the optimal approximation problem is obtained.     

An inverse eigenvalue problem for periodic Jacobi matrices in Minkowski spaces

The spectral properties of periodic Jacobi matrices in Minkowski spaces are studied. An inverse problem for these matrices is investigated, and necessary and sufficient conditions under which the problem is solvable are presented. Uniqueness results are also discussed, and an algorithm to construct the solutions and illustrative examples is provided.     

An inverse eigenvalue problem and an associated approximation problem for generalized K-centrohermitian matrices

A partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized K-centrohermitian matrices are considered. It is shown under which conditions the inverse eigenproblem has a solution. An expression of its general solution is given. In case a solution of the inverse eigenproblem exists, the optimal approximation problem can be solved. The formula of its unique solution is given.     

Direct and inverse spectral problems for a class of non-self-adjoint periodic tridiagonal matrices

The spectral properties of a class of tridiagonal matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The obtained results extend some results on the direct and inverse spectral problems for periodic Jacobi matrices and for some non-self-adjoint tridiagonal matrices.     

A Jacobi matrix inverse eigenvalue problem with mixed data

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from part of its eigenvalues and its leading principal 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.     

Solutions to an inverse monic quadratic eigenvalue problem

Given n+1 pairs of complex numbers and vectors (closed under complex conjugation), the inverse quadratic eigenvalue problem is to construct real symmetric or anti-symmetric matrix C and real symmetric matrix K of size n×n so that the quadratic pencil Q(λ)=λ²I_n+λC+K has the given n+1 pairs as eigenpairs. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained. Numerical algorithms for solving the problem are developed. Numerical examples illustrating these solutions are presented.     

A generalized inverse eigenvalue problem in structural dynamic model updating

This paper is concerned with the problem of the best approximation for a given matrix pencil under a given spectral constraint and a submatrix pencil constraint. Such a problem arises in structural dynamic model updating. By using the Moore–Penrose generalized inverse and the singular value decomposition (SVD) matrices, the solvability condition and the expression for the solution of the problem are presented. A numerical algorithm for solving the problem is developed.     

Inverse eigenvalue problems for linear complementarity systems

Traditionally an inverse eigenvalue problem is about reconstructing a matrix from a given spectral data. In this work we study the set of real matrices A of order n such that the linear complementarity system

     

Augmented Block Householder Arnoldi Method

Computing the eigenvalues and eigenvectors of a large sparse nonsymmetric matrix arises in many applications and can be a very computationally challenging problem. In this paper we propose the Augmented Block Householder Arnoldi (ABHA) method that combines the advantages of a block routine with an augmented Krylov routine. A public domain MATLAB code ahbeigs has been developed and numerical experiments indicate that the code is competitive with other publicly available codes.     

Solving inverse cone-constrained eigenvalue problems

We compare various algorithms for constructing a matrix of order $n$ whose Pareto spectrum contains a prescribed set $\Lambda =\{\lambda _1,\ldots , \lambda _p\}$ of reals. In order to avoid overdetermination one assumes that $p$ does not exceed $n^2.$ The inverse Pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations. We also address the issue of computing Lorentz spectra and solving inverse Lorentz eigenvalue problems.     

Fast accurate eigenvalue methods for graded positive definite matrices

Summary. Let where is a positive definite matrix and is diagonal and nonsingular. We show that if the condition number of is much less than that of then we can use algorithms based on the Cholesky factorization of to compute the eigenvalues of to high relative accuracy more efficiently than by Jacobi's method. The new methods are generally slower than tridiagonalization methods (which do not deliver the eigenvalues to maximal relative accuracy) but can be up to 4 times faster when the condition number of is very large. Received April 13, 1995     

The eigenvalue distribution of products of Toeplitz matrices - Clustering and attraction

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.     

Transition matrices for well-conditioned Markov chains

Let T∈R^n×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where A_j, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ₃ provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ₃ is as small as possible.     

An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices

We express the eigenvalues of a pentadiagonal symmetric Toeplitz matrix as the zeros of explicitly given rational functions.