Inversion of polynomial and rational matrices

The inversion of polynomial and rational matrices is considered. For regular matrices, three algorithms for computing the inverse matrix in a factored form are proposed. For singular matrices, algorithms of constructing pseudoinverse matrices are considered. The algorithms of inversion of rational matrices are based on the minimal factorization which reduces the problem to the inversion of polynomial matrices. A class of special polynomial matrices is regarded whose inverse matrices are also polynomial matrices. Inversion algorithms are applied to the solution of systems with polynomial and rational matrices. Bibliography: 3 titles. Translated by V. N. Kublanovskaya. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 97–109.     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices

The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity.     

多项式矩阵代数性质讨论

本文利用系统与控制论中有关多项式矩阵的结果,对多项式矩阵代数性质进行讨论,得到的主要结果有多项式方阵环是主理想环,也是主单侧理想环。     

Polynomial Fibonacci–Hessenberg matrices

A Fibonacci–Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci–Hessenberg matrix. Several classes of polynomial Fibonacci–Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci–Hessenberg matrices satisfying this property are given.     

To Solving Multiparameter Problems of Algebra. 4. The AB-Algorithm and its Applications

The paper continues the investigation of methods for factorizing q-parameter polynomial matrices and considers their applications to solving multiparameter problems of algebra. An extension of the AB-algorithm, suggested earlier as a method for solving spectral problems for matrix pencils of the form A - λB, to the case of q-parameter (q ≥ 1) polynomial matrices of full rank is proposed. In accordance with the AB-algorithm, a finite sequence of q-parameter polynomial matrices such that every subsequent matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain rule for selecting specific basis matrices is described. Applications of the AB-algorithm to computing complete polynomials of a q-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial solutions of an arbitrary q-parameter polynomial matrix are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 127–143.     

To the Solution of Partial Eigenproblems for Polynomial Matrices

Methods for computing scalar and vector spectral characteristics of a polynomial matrix are proposed. These methods are based on determining the so-called generating vectors (eigenvectors and principal vectors) by using the method of rank factorization of polynomial matrices. The possibility of extending the methods to the case of two-parameter polynomial matrices is indicated. Bibliography: 4 titles.     

矩阵多项式的平方根矩阵

研究了矩阵多项式的开平方问题,给出了矩阵多项式能开平方的充分必要条件及其平方根矩阵的个数,包含并推广了文[1]中的主要结论.     

On some factorizations of two-parameter polynomial matrices

This paper is a logical continuation of the author's discussion about the solution of spectral problems for two-parameter polynomial matrices of general type. Various rank factorization algorithms are suggested, among them the so-called minimal factorization of a singular two-parameter polynomial matrix of degenerate rank into a product of some matrices whose ranks are equal to the rank of the original matrix. Spectral properties of these matrices are studied. The notion of minimal factorization is also extended to one-parameter polynomial and constant matrices. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 94–116 Translated by V. N. Kublanovskaya.     

To Solving Multiparameter Problems of Algebra. 5. The ∇V-q Factorization Algorithm and its Applications

The algorithm of ∇V-factorization, suggested earlier for decomposing one- and two-parameter polynomial matrices of full row rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix, whereas the regular spectrum is empty), is extended to the case of q-parameter (q ≥ 1) polynomial matrices. The algorithm of ∇V-q factorization is described, and its justification and properties for matrices with arbitrary number of parameters are presented. Applications of the algorithm to computing irreducible factorizations of q-parameter matrices, to determining a free basis of the null-space of polynomial solutions of a matrix, and to finding matrix divisors corresponding to divisors of its characteristic polynomial are considered. Bibliogrhaphy: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 144–153.     

Displacement structures and fast inversion formulas for confluent polynomial Vandermonde-like matrices

In the present paper, confluent polynomial Vandermonde-like matrices with general recurrence structure are introduced. Three kinds of displacement structure equations and two kinds of fast inversion formulas for this class of matrices are derived by using displacement structure matrix method. A relationship between confluent polynomial Vandermonde-like matrices and confluent Cauchy-like matrices is pointed out.     

Containment regions for zeros of polynomials from numerical ranges of companion matrices

Containment regions for the zeros of a monic polynomial are given with the aid of results for containment regions for the numerical range of certain bordered diagonal matrices which are applied to different types of companion matrices of the polynomial.     

Generalized Leibniz functional matrices and factorizations of some well-known matrices

In this paper, we introduce the generalized Leibniz functional matrices and study some algebraic properties of such matrices. To demonstrate applications of these properties, we derive several novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix. In addition, by applying factorizations of the generalized Leibniz functional matrices, we redevelop the known results of factorizations of Stirling matrices of the first and second kind and the generalized Pascal matrix.     

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.     

Spectral factorization of non-symmetric polynomial matrices

The topic of the paper is spectral factorization of rectangular and possibly non-full-rank polynomial matrices. To each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients. The associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix. The matrix dimensions of the pencil we obtain by solving an integer linear programming (ILP) minimization problem. Then by extracting a deflating subspace of the pencil we come to the required spectral factorization. We apply the algorithm to most general-case of inner–outer factorization, regardless continuous or discrete time case, and to finding the greatest common divisor of polynomial matrices.     

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.     

Semiscalar equivalence and the smith normal form of polynomial matrices

We give conditions under which a set of polynomial matrices over a finite field can be simultaneously reduced by means of semiscalar equivalent transformations to a special triangular form with invariant factors on the principal diagonals. We investigate multiplicative properties of the Smith normal form of polynomial matrices and in particular we identify a class of polynomial matrices for which the Smith normal form of the product matrix is equal to the product of the Smith normal forms of the factor matrices.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 13–16, 1987.     

The least-squares method for matrices dependent on parameters

Some algorithms are suggested for constructing pseudoinverse matrices and for solving systems with rectangular matrices whose entries depend on a parameter in polynomial and rational ways. The cases of one- and two-parameter matrices are considered. The construction of pseudoinverse matrices are realized on the basis of rank factorization algorithms. In the case of matrices with polynomial occurrence of parameters, these algorithms are the ΔW-1 and ΔW-2 algorithms for one- and two-parameter matrices, respectively. In the case of matrices with rational occurrence of parameters, these algorithms are the irreducible factorization algorithms. This paper is a continuation of the author's studies of the solution of systems with one-parameter matrices and an extension of the results to the case of two-parameter matrices with polynomial and rational entries. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 176–185. This work was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919). Translated by V. N. Kublanovskaya.     

To Solving Multiparameter Problems of Algebra. I. Methods for Computing Complete Polynomials and Their Applications

The notion of a complete polynomial of a multiparameter polynomial matrix, generalizing that of an invariant polynomial of a one-parameter polynomial matrix, is introduced. Methods for computing complete polynomials are suggested. Applications of these methods to various spectral problems for polynomial matrices are considered. Bibliography: 7 titles.     

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.