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.     

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.     

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.     

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 backward error for the inverse singular value problem

A backward error for inverse singular value problems with respect to an approximate solution is defined, and an explicit expression for the backward error is derived by extending the approach described in [J.G. Sun, Backward errors for the inverse eigenvalue problem, Numer. Math. 82 (1999) 339-349]. The expression may be useful for testing the stability of practical algorithms.     

An algorithm for determining copositive matrices

In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of on the standard simplex Δ_m, where each component of the vector β is −1, 0 or 1.     

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.     

Backward errors for eigenproblem of two kinds of structured matrices

In this paper, we consider backward errors in the eigenproblem of symmetric centrosymmetric and symmetric skew-centrosymmetric matrices. By making use of the properties of symmetric centrosymmetric and symmetric skew-centrosymmetric matrices, we derive explicit formulae for the backward errors of approximate eigenpairs.     

HGAH矩阵的逆特征值问题

In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamihonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.     

An algorithm for addressing the real interval eigenvalue problem

In this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices. Such matrices could be used to model real-life problems, where data sets suffer from bounded variations such as uncertainties (e.g. tolerances on parameters, measurement errors), or to study problems for given states.The algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis. The quality of the computed approximation and the running time of the algorithm depend on a given input accuracy. We also present an efficient C++ implementation and illustrate its efficiency on various data sets. In most of the cases we manage to compute efficiently the exact boundary points (limited by floating point representation).     

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.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.     

On perturbation bounds of generalized eigenvalues for diagonalizable pairs

The purpose of this paper is to study the perturbation of generalized eigenvalues. Two perturbation bounds of the diagonalizable pairs are given. These results extend the corresponding ones given by Sun (Math Numer Sinica 4:23–29, 1982). This work is supported by the Natural Science Foundation of Guangdong Province (No. 06025061) and by the National Natural Science Foundations of China (No. 10671077 and 10626021).     

Current inverse iteration software can fail

Inverse iteration is widley used to compute the eigenvectors of a matrix once accurate eigenvalues are known. We discuss various issues involved in any implementation of inverse iteration for real, symmetric matrices. Current implementations resort to reorthogonalization when eigenvalues agree to more than three digits relative to the norm. Such reorthogonalization can have unexpected consequences. Indeed, as we show in this paper, the implementations in EISPACK and LAPACK may fail. We illustrate with both theoretical and empirical failures. This research was supported, while the author was at the University of California, Berkeley, in part by DARPA Contract No. DAAL03-91-C-0047 through a subcontract with the University of Tennessee, DOE Contract No. DOE-W-31-109-Eng-38 through a subcontract with Argonne National Laboratory, by DOE Grant No. DE-FG03-94ER25219, NSF Grant Nos. ASC-9313958 and CDA-9401156, and DOE Contract DE-AC06-76RLO 1830 through the Environmental Molecular Sciences construction project at Pacific Northwest National Laboraotry (PNNL).     

The inverse eigenvalue problem of generalized reflexive matrices and its approximation

This paper studies inverse eigenvalue problems of generalized reflexive matrices and their optimal approximations. Necessary and sufficient conditions for the solvability of the problems are derived, the solutions and their optimal approximations are provided.     

H-矩阵和S-双对角占优矩阵

In this paper, the concept of the s-doubly diagonally dominant matrices is introduced and the properties of these matrices are discussed. With the properties of the s-doubly diagonally dominant matrices and the properties of comparison matrices, some equivalent conditions for H-matrices are presented. These conditions generalize and improve existing results about the equivalent conditions for H-matrices. Applications and examples using these new equivalent conditions are also presented, and a new inclusion region of k-multiple eigenvalues of matrices is obtained.     

Exact eigensystems for some matrices arising from discretizations

It is known, for example, that the eigenvalues of the N×N matrix A, arising in the discretization of the wave equation, whose only nonzero entries are A_kk+1=A_k+1k=-1,k=1,…,N-1, and A_kk=2,k=1,…,N, are 2{1-cos[pπ/(N+1)]} with corresponding eigenvectors v^(p) given by . We show by considering a simple finite difference approximation to the second derivative and using the summation formulae for sines and cosines that these and other similar formulae arise in a simple and unified way.