On inverse multiplicative eigenvalue problems for matrices

Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ₁,…,λ_n} in C. The number of different matrices D is at most n!.     

Solving inverse eigenvalue problems via Householder and rank-one matrices

A brief and practical algorithm is introduced to solve symmetric inverse eigenvalue problems, which we call HROU algorithm. The algorithm is based on Householder transformations and rank one updating. We give some basic properties and the computational amount and develop sensitivity analysis of HROU algorithm. Furthermore, we develop HROU algorithm into a multi-level and adaptive one, named MLAHROU, to solve symmetric nonnegative inverse eigenvalue problems. New sufficient conditions to ensure symmetric nonnegative matrices and symmetric M-matrices are given. Many numerical examples are given to verify our theory, compare with existing results and show the efficiency of our algorithms.     

To solving spectral problems for multiparameter polynomial matrices

An approach to solving spectral problems for multiparameter polynomial matrices based on passing to accompanying pencils of matrices is described. Also reduction of spectral problems for multiparameter pencils of complex matrices to the corresponding real problems is considered. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2006, pp. 212–231.     

The solvability conditions for the inverse eigenvalue problems of reflexive matrices

This paper involves related inverse eigenvalue problems of reflexive matrices and their optimal approximation, the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue, and the general provided form of the solution. Furthermore, the algorithm to compute the optimal approximate solution and some numerical experiments are given.     

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.     

Large a quasi-Jacobi form for J-normal matrices and inverse eigenvalue problems

A quasi-Jacobi form for J-unitarily diagonalizable J-normal matrices is given, extending a result for normal matrices due to Malamud. The inverse eigenvalue problem for J-normal matrices satisfying certain prescribed spectral conditions is investigated. It is shown that there exists unicity in the case of pseudo-Jacobi matrices.     

To solving problems of algebra for two-parameter matrices. VII

The paper discusses the method of rank factorization for solving spectral problems for two-parameter polynomial matrices. New forms of rank factorization, which are computed using unimodular matrices only, are suggested. Applications of these factorizations to solving spectral problems for two-parameter polynomial matrices of both general and special forms are presented. In particular, matrices free of the singular spectrum are considered. Conditions sufficient for a matrix to be free of the singular spectrum and also conditions sufficient for a basis matrix of the null-space to have neither the finite regular nor the finite singular spectrum are provided. Bibliography: 3 titles.     

To solving problems of algebra for two-parameter matrices. II

This paper is one of the series of survey papers dedicated to the development of methods for solving problems of algebra for two-parameter polynomial matrices of general form. The paper considers the AB-algorithm and the ∇V-2 factorization algorithm, together with their applications. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 150–165.     

To solving problems of algebra for two-parameter matrices. VIII

The paper discusses the method of hereditary pencils for computing points of the regular and singular spectra of a general two-parameter polynomial matrix. The method allows one to reduce the spectral problems considered to eigenproblems for polynomial matrices and pencils of constant matrices. Algorithms realizing the method are suggested and justified. Bibliography: 4 titles.     

To solving problems of algebra for two-parameter matrices. IV

This paper continues the series of publications devoted to surveying and developing methods for solving the following problems for a two-parameter matrix F (λ, μ) of general form: exhausting points of the mixed regular spectrum of F (λ, μ); performing operations on polynomials in two variables (computing the GCD and LCM of a few polynomials, division of polynomials, and factorization); computing a minimal basis of the null-space of polynomial solutions of the matrix F (λ, μ) and separation of its regular kernel; inversion and pseudo in version of polynomial and rational matrices in two variables, and solution of systems of nonlinear algebraic equations in two unknowns. Most of the methods suggested are based on rank factorizations of a two-parameter polynomial matrix and on the method of hereditary pencils. Bibliography: 8 titles.     

To solving problems of algebra for two-parameter matrices. III

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem of solving an equation F(λ, μ)x = 0 with a polynomial two-parameter matrix F(λ, μ) to solving an equation of the form D(λ, μ)y = 0, where D(λ, μ) = A(μ)-λB(μ) is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix F(λ, μ) are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 166–207.     

To solving problems of algebra for two-parameter matrices. I

This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations and, in particular, the relatively irreducible and ΔW-2 factorizations, which are used in solving spectral problems for two-parameter polynomial matrices F(λ, μ). Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of F(λ, μ). The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of F(λ, μ) are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the ΔW-2 method are studied. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 107–149.     

To solving problems of algebra for two-parameter matrices. VI

The paper continues the series of papers devoted to surveying and developing methods for solving problems for two-parameter polynomial and rational matrices. Different types of factorizations of two-parameter rational matrices (including irreducible and minimal ones), methods for computing them, and their applications to solving spectral problems are considered. Bibliography: 6 titles.