Common multiples and common divisors of matrix polynomials, II. Vandermonde and resultant matrices

In this paper explicit formulas are given for least common multiples and greatest common divisors of a finite number of matrix polynomials in terms of the coefficients of the given polynomials. An important role is played by block matrix generalizations of the classical Vandermonde and resultant matrices. Special attention is given to the evaluation of the degrees and other characteristics. Applications to matrix polynomial equations and factorization problems are made.     

Parallel factorizations of matrix polynomials over an arbitrary field

We study the problem of the decomposition of a matrix polynomial over an arbitrary field into a product of factors of lower degrees with preassigned characteristic polynomials. We find necessary conditions for the existence of the required factorization, which are also sufficient for certain classes of matrix polynomials. The proposed method makes it possible to solve the problem completely for matrix polynomials with one nonconstant invariant factor. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

A new type of Hermite matrix polynomial series

Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new generating functions for Hermite matrix polynomials and examines existence and convergence of their associated series expansion by using Mehler’s formula for the general matrix case. Moreover, we derive interesting new relations for even- and odd-power summation in the generating-function expansion containing Hermite matrix polynomials. Some new results for the scalar case are also presented.     

Rectangular matrix Padé approximants and square matrix orthogonal polynomials

In this paper we study Padé-type and Padé approximants for rectangular matrix formal power series, as well as the formal orthogonal polynomials which are a consequence of the definition of these matrix Padé approximants. Recurrence relations are given along a diagonal or two adjacent diagonals of the table of orthogonal polynomials and their adjacent ones. A matrix qd-algorithm is deduced from these relations. Recurrence relations are also proved for the associated polynomials. Finally a short presentation of right matrix Padé approximants gives a link between the degrees of orthogonal polynomials in right and left matrix Padé approximants in order to show that the latter are identical. This revised version was published online in June 2006 with corrections to the Cover Date.     

Isotropic covariance matrix polynomials on spheres

This article characterizes the covariance matrix function of a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the sphere. By applying the characterization to examine the validity of a matrix function whose entries are polynomials of degrees up to 4, we obtain a necessary and sufficient condition for the polynomial matrix to be an isotropic covariance matrix function on the sphere.     

Nevanlinna Matrices of Entire Functions

The notion of a pre-Nevanlinna matrix of entire functions is introduced, and we find necessary and sufficient conditions for an entire function to belong to such a matrix, thereby generalizing previous work of Krein. If one of the functions in a pre-Nevanlinna matrix is a polynomial, then the three others arc also polynomials and their degrees differ by at most two. If the functions in a pre-Nevanlinna matrix are transcendental they have necessarily the same order, type and indicators.     

On the characteristic and Laplacian polynomials of trees

We find the characteristic polynomials of adjacency and Laplacian matrices of arbitrary unweighted rooted trees in term of vertex degrees, using the concept of the rooted product of graphs. Our result generalizes a result of Rojo and Soto [O. Rojo, R. Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl. 403 (2005) 97–117] on a special class of rooted unweighted trees, namely the trees such that their vertices in the same level have equal degrees.     

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship. R.S. was supported in part by National Science Foundation grant DMS-9988459.     

Matrix Padé Approximation of rational functions

In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the smallest degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table.     

A new extension of Hermite matrix polynomials and its applications

In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced.     

Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.     

A type of matrix Padé approximant inspired by scalar component models

In this paper we define a type of matrix Padé approximant inspired by the identification stage of multivariate time series models considering scalar component models. Of course, the formalization of certain properties in the matrix Padé approximation framework can be applied to time series models and in other fields. Specifically, we want to study matrix Padé approximants as follows: to find rational representations (or rational approximations) of a matrix formal power series, with both matrix polynomials, numerator and denominator, satisfying three conditions: (a) minimum row degrees for the numerator and denominator, (b) an invertible denominator at the origin, and (c) canonical representation (without free parameters).     

Computing orthogonal polynomials in Sobolev spaces

Summary. Numerical methods are considered for generating polynomials orthogonal with respect to an inner product of Sobolev type, i.e., one that involves derivatives up to some given order, each having its own (positive) measure associated with it. The principal objective is to compute the coefficients in the increasing-order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next. As a by-product of this computation, one gains access to the zeros of these polynomials via eigenvalues of an upper Hessenberg matrix formed by the coefficients generated. Two methods are developed: One is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as "modified Chebyshev algorithm". The other - a generalization of "Stieltjes's procedure" - expresses the desired coefficients in terms of a Sobolev inner product involving the orthogonal polynomials in question, whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point. The numerical characteristics of these methods are illustrated in the case of Sobolev orthogonal polynomials of old as well as new types. Based on extensive numerical experimentation, a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros. Received July 13, 1994 / Revised version received September 26, 1994     

Upward Extension of the Jacobi Matrix for Orthogonal Polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

A Capelli type theorem for multiplicative convolutions of polynomials

We prove a Capelli type theorem on the canonical decomposition for multiplicative convolutions of polynomials. We derive then some irreducibility criteria for convolutions of polynomials in several variables over a given field. The irreducibility conditions are expressed only in terms of the degrees of the polynomials in convolution, the degrees of their coefficients, and the degrees of some suitable divisors of the resulting leading coefficient. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于矩阵多项式特征值界的注记

本文讨论矩阵多项式特征值定域问题.首先对Higham和Tisseur[Linear Algebra Appl.,358(2003),5-22]得到的结果给出较详细的比较.然后利用分块矩阵谱半径的估计给出了获取特征值界的一种新办法.利用这种新办法,不但可以简明地得出很多已有的界,且对椭圆及双曲矩阵多项式得出了特征值的新的界.     

The Bezoutian and the eigenvalue-separation problem for matrix polynomials

A generalized Bezout matrix for a pair of matrix polynomials is studied and, in particular, the structure of its kernel is described and the relations to the greatest common divisor of the given matrix polynomials are presented. The classical root-separation problems of Hermite, Routh-Hurwitz and Schur-Cohn are solved for matrix polynomials in terms of this Bezout matrix. The eigenvalue-separation results are also expressed in terms of Hankel matrices whose entries are Markov parameters of rational matrix function. Some applications of Jacobi's method to these problems are pointed out.     

Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification

The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient and necessary condition for a quasihyperbolic matrix polynomial to be strictly isospectral to a real diagonal quasihyperbolic matrix polynomial of the same degree, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.     

Relative asymptotics for orthogonal matrix polynomials

In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced.