Random Block Matrices and Matrix Orthogonal Polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.     

Eigenvalues of Toeplitz matrices associated with orthogonal polynomials

A connection between the asymptotic distribution of the zeros of orthogonal polynomials and the asymptotic behavior of the eigenvalues of Toeplitz matrices associated with these orthogonal polynomials is given. The result is applied to various families of orthogonal polynomials.     

Circular β ensembles,CMV representation,characteristic polynomials

In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where β > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle. This work was supported by National Natural Science Foundation of China (Grant No. 10671176)     

The spread of matrices and polynomials

The spread of a matrix (or polynomial) is the maximum distance between any two of its eigenvalues (or its zeros). E. Deutsch has recently given upper bounds for the spread of matrices and polynomials. We obtain sharper, simpler upper bounds and observe that they are also upper bounds for the sum of the absolute values of the two largest eigenvalues (or zeros).     

On the spread of matrices and polynomials

The spread of a matrix (polynomial) is defined as the maximum distance between two of its eigenvalues (zeros). In this note upper bounds are found for the spread of matrices and polynomials.     

Interpretations of some parameter dependent generalizations of classical matrix ensembles

Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Andersons and Dixons work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.Supported by the Australian Research Council     

Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter

A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.     

Multiple Meixner–Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner–Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest.The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner–Pollaczek polynomials arise in an inhomogeneous version of this model.     

Estimates for the real and imaginary parts of the eigenvalues of matrices and applications

This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

A note on biorthogonal ensembles

We study multiple orthogonal polynomials in the context of biorthogonal ensembles of random matrices. In these ensembles, the eigenvalue probability density function factorizes into a product of two determinants while the eigenvalue correlation functions can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term, whose multiple polynomials are related to the modified Bessel function of the first kind.     

Asymptotics of Random Density Matrices

We investigate random density matrices obtained by partial tracing larger random pure states. We show that there is a strong connection between these random density matrices and the Wishart ensemble of random matrix theory. We provide asymptotic results on the behavior of the eigenvalues of random density matrices, including convergence of the empirical spectral measure. We also study the largest eigenvalue (almost sure convergence and fluctuations). Submitted: February 9, 2007. Accepted: March 3, 2007.     

The Bezoutian and the algebraic Riccati equation

The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.     

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.     

The Bezoutian and the algebraic Riccati equation

The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.     

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.     

Bounds and Majorization Relations for the Zeros of a Class of Fibonacci-Type Polynomials

We apply certain matrix inequalities involving eigenvalues, the numerical radius, and the spectral radius to obtain new bounds and majorization relations for the zeros of a class of Fibonacci-type polynomials. Our results improve upon some earlier bounds for the zeros of these polynomials.     

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     

Error analysis of signal zeros from a related companion matrix eigenvalue problem

An error analysis of the so-called signal zeros of polynomials linked to exponentially damped signals is performed and error bounds are derived. The analysis uses the link between polynomials and companion matrices and allows us to show that the related companion matrix eigenvalue problem is governed by the condition number of a rectangular Vandermonde matrix which has the zeros of interest as nodes. Conditions under which the zeros are well conditioned are discussed.     

The asymptotical spectrum of Jacobi matrices

A method to calculate the asymptotical eigenvalue density (asymptotical density of zeros) ρ(x) of Jacobi matrices (orthogonal polynomials) in terms of its moments is presented. This method does not require the convergence of continued fractions and inversion of functional transformations as previous ones do. It is shown to be applicable to a wide family of Jacobi matrices (orthogonal polynomials). As a byproduct the density ρ(x) is explicitly found for certain classical orthogonal polynomials.