Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices

Vandermonde matrices with real nodes are known to be severely ill-conditioned. We investigate numerically the extent to which the condition number of such matrices can be reduced, either by row-scaling or by optimal configurations of nodes. In the latter case we find empirically the condition of the optimally conditioned n×n Vandermonde matrix to grow exponentially at a rate slightly less than \((1+\sqrt{2})^{n}\). Much slower growth—essentially linear—is observed for optimally conditioned Vandermonde-Jacobi matrices. We also comment on the computational challenges involved in determining condition numbers of highly ill-conditioned matrices.     

How bad are Hankel matrices?

Summary. Considered are Hankel, Vandermonde, and Krylov basis matrices. It is proved that for any real positive definite Hankel matrix of order , its spectral condition number is bounded from below by . Also proved is that the spectral condition number of a Krylov basis matrix is bounded from below by . For , a Vandermonde matrix with arbitrary but pairwise distinct nodes , we show that ; if either or for all , then . Received January 24, 1993/Revised version received July 19, 1993     

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.     

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.     

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.     

To solving multiparameter problems of algebra. 9. The Ψ<Emphasis Type="Italic">F-q</Emphasis> method for factorizing invariant polynomials and its applications

A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173.     

An approach to solving multiparameter algebraic problems

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246. Translated by V. N. Kublanovskaya.     

The {\cal {D Q R}} algorithm, basic theory, convergence, and conditional stability

Summary. We describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on –orthogonal reduction techniques where is computed from the given matrix . It operates on a similarity reduction of a real matrix to general tridiagonal form and computes all of 's eigenvalues in operations, where the part of the operations is possibly performed over , instead of the 7–8 real flops required by the eigenvalue algorithm. Potential breakdo wn of the algorithm can occur in the reduction to tridiagonal form and in the –orthogonal reductions. Both, however, can be monitored during the computations. The former occurs rather rarely for dimensions and can essentially be bypassed, while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the steps. We prove an implicit theorem which allows implicit shifts, give a convergence proof for the algorithm and show that is conditionally stable for general balanced tridiagonal matrices . Received April 25, 1995 / Revised version received February 9, 1996     

Biorthogonal Polynomial Bases and Vandermonde-like Matrices

This article considers a family of Gram matrices of pairs of bases of a finite dimensional vector space of polynomials with respect to certain indefinite inner products. Such a family includes all the generalized confluent Vandermonde matrices relative to any polynomial basis, like the Chebyshev-Vandermonde matrices, for example. Using the biorthogonality of pairs of bases with respect to a divided difference functional, properties of matrices and functionals, as well as interpolation formulas are obtained. I show that the computation of the inverse of a Vandermonde-like matrix is essentially equivalent to the computation of the partial fractions decompositions of a set of rational functions with a common denominator. I also explain why the various Chebyshev-Vandermonde matrices are the simplest generalizations of the classic Vandermonde matrices and describe a simple algorithm for the computation of their inverses, which requires a number of multiplications of the order of 3N².     

Hermite matrix polynomials and second order matrix differential equations

In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y″−xAY′+BY=0. An explicit expression for the Hermite matrix polynomials, the orthogonality property and a Rodrigues’ formula are given.     

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.     

Multivariate Polynomials,Duality, and Structured Matrices

We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues, and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians, and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to the matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.     

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.     

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (s_j)^∞_{j = 0} of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case.     

The conformal ‘bratwurst’ mapsand associated Faber polynomials

Summary. Conformal maps from the exterior of the closed unit disk onto the exterior of ‘bratwurst’ shape sets in the complex plane are constructed. Using these maps, coefficients for the computation of the corresponding Faber polynomials are derived. A ‘bratwurst’ shape set is the result of deforming an ellipse with foci on the real axis, by conformally mapping the real axis onto the unit circle. Such sets are well suited to serve as inclusion sets for sets associated with a matrix, for example the spectrum, field of values or a pseudospectrum. Hence, the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems. The main advantage of the approach is that the conformal maps are derived from elementary transformations, allowing an easy computation of the associated transfinite diameter, asymptotic convergence factor and Faber polynomials. Numerical examples are given. Received October 7, 1998 / Revised version received March 15, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.     

The mathematics of eigenvalue optimization

 Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread. Received: December 4, 2002 / Accepted: April 22, 2003 Published online: May 28, 2003 Key Words.  eigenvalue optimization – convexity – nonsmooth analysis – duality – semidefinite program – subdifferential – Clarke regular – chain rule – sensitivity – eigenvalue perturbation – partly smooth – spectral function – unitarily invariant norm – hyperbolic polynomial – stability – robust control – pseudospectrum – H _∞ norm Mathematics Subject Classification (2000): 90C30, 15A42, 65F15, 49K40     

S-matrices

A new class of so called S-matrices is introduced which allows investigating links between various known classes of matrices such as Vandermonde matrices, Hankel matrices, companion matrices, etc. For complex S-matrices, the problem of decomposition into a quasidirect sum (a sum for which the sum of the ranks of the summands equals the rank of the given matrix) of indecomposable complex S-matrices is completely solved, and the uniqueness of such a decomposition is proved.     

To solving multiparameter problems of algebra. 8. The <Emphasis Type="Italic">RP-q</Emphasis> method and its applications

A new method (the RP-q method) for factorizing scalar polynomials in q variables and q-parameter polynomial matrices (q ≥ 1) of full rank is suggested. Applications of the algorithm to solving systems of nonlinear algebraic equations and some spectral problems for a q-parameter polynomial matrix F (such as separation of the eigenspectrum and mixed spectrum of F, computation of bases with prescribed spectral properties of the null-space of polynomial solutions of F, and computation of the hereditary polynomials of F) are considered. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 149–164.     

Maximal semigroups dominated by 0–1 matrices

Maximal semigroups dominated by a 0–1 matrix of a certain type are determined. The 0–1 matrices that dominate maximal bounded and maximal commuting semigroups are given. Also semigroup modules over maximal semigroups dominated by a 0–1 matrix are discussed.