On inversion of hermitian block toeplitz matrices

A formula for the inverse of a hermitian block Toeplitz matrix via solutions of two block equations(instead of four in the general case) is given.     

Block toeplitz products of block toeplitz matrices

Necessary and sufficient conditions for the product of two block Toeplitz matrices to be block Toeplitz are obtained. In the special case of two Toeplitz matrices, the conditions simplify considerably and, when combined with known necessary and sufficient conditions for a nonsingular Toeplitz matrix to have a Toeplitz inverse, provide a simple characterization of the additional matrix structure required by a subclass of Toeplitz matrices in order for it to be closed with respect to both inversion and multiplication.     

On the eigenvalues of matrices

Summary Let A be an n×n matric with arbitrary complex elements and with eigen-values λ₁, λ₂, ..., λ_n. A method is described for the approximàte determination of max | λ_j | ; characteristical is that prescribing a percentual error the number of elementary operations of the process, necessary to reach such precision, depends only on n and not on the elements of A More general characteristical equations are also considered. To Enrico Bompiani on his scientific Jubilee     

Infinite analogues of block toeplitz matrices and related orthogonal functions

This paper concerns a class of infinite block matrices that are analogous to finite block Toeplitz matrices. Also studied are corresponding matrix-valued functions that are orthogonal for a matrixvalued inner product. An appendix presents basic results on orthogonalization in a Hilbert module.     

On invertibility of finite sections of toeplitz matrices

AMS(MOS) # 47B 35 15A 09     

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.     

On relative perturbation theory for eigenvalues and eigenvectors of block operator matrices

We are concerned with singularly perturbed spectral problems which appear in engineering sciences. Typically under the influence of a singular perturbation the model can be approximated by a simpler, perturbation independent model. Such reduced model is usually better amenable to analytic or numeric analysis. However, the question of the quality of approximation has to be answered. Frequently, correctors which yield an improved solution–capturing important phenomena which the reduced model does not “see”–to the original problems are required. We tackle both question for self-adjoint eigenvalue/eigenvector problems posed in a general Hilbert space. Our technique is constructive and is based on methods (relative perturbation theory) of modern Numerical Linear Algebra. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large deviations of the extreme eigenvalues of random deformations of matrices

Consider a real diagonal deterministic matrix X _n of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale n, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of X _n converge to the edges of the support of the limiting measure and when we allow some eigenvalues of X _n , that we call outliers, to converge out of the bulk. We can also generalise our results to the case when X _n is random, with law proportional to e ^{?n Tr V(X)} dX, for V growing fast enough at infinity and any perturbation of finite rank.     

On eigenvalues of rectangular matrices

Given a (k+1)-tuple A,B ₁, ..., B _k of m×n matrices with m ≤ n, we call the set of all k-tuples of complex numbers {λ ₁, ..., λ k} such that the linear combination A+λ ₁ B ₁+λ ₂ B ₂+ ... +λ _k B _k has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications to multiparameter generalizations of the Heine-Stieltjes spectral problem, we study a number of properties of the eigenvalue locus in the most important case k = n−m+1.     

On determinants of nonprincipal submatrices of hermitian matrices

It is our purpose to compute the maximum value of the modulus of the determinant of an m×m nonprincipal submatrix of an n×n hermitian (or real symmetric) matrix A, in terms of m, the eigenvalues of A, and the cardinality k of the set of common row and column indices of this submatrix.     

On the product of the diagonal elements of hermitian matrices

In [5] M. Marcus stated a conjecture concerning the product of the diagonal elements of normal matrices which is false [6]. In this paper we prove that such a conjecture is true in the Hermitian case.     

An algorithm for computing the eigenvalues of block companion matrices

In this paper we propose a method for computing the roots of a monic matrix polynomial. To this end we compute the eigenvalues of the corresponding block companion matrix C. This is done by implementing the QR algorithm in such a way that it exploits the rank structure of the matrix. Because of this structure, we can represent the matrix in Givens-weight representation. A similar method as in Chandrasekaran et al. (Oper Theory Adv Appl 179:111–143, 2007), the bulge chasing, is used during the QR iteration. For practical usage, matrix C has to be brought in Hessenberg form before the QR iteration starts. During the QR iteration and the transformation to Hessenberg form, the property of the matrix being unitary plus low rank numerically deteriorates. A method to restore this property is used.     

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

We consider banded block Toeplitz matrices T_n with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of T_n for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of T_n and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if T_n is a Hessenberg matrix.     

On the eigenvalues of non-negative Jacobi matrices

The spectrum σ of a non-negative Jacobi matrix J is characterized. If J is also required to be irreducible, further conditions on σ are needed, some of which are explored.     

On the eigenvalues of double band matrices

We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form rζ, where r is a nonnegative real number and ζ is a pth root of unity, where p is the period of the matrix, which is computed from the distance between the bands. We also present a problem in the asymptotics of spectra in which such double band matrices are perturbed by banded matrices.     

On the eigenvalues of some tridiagonal matrices

A solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by William Trench. The method presented can be generalizable to other problems.     

On the eigenvalues of some transfer matrices

We consider the roots of two families of polynomials which can be derived as the characteristic polynomials of some (generalized) transfer matrices. We study the possible multiplicities and the number of real roots. Moreover, the number of roots lying inside the unit disk is determined, and bounds for their modulus and for the modulus of the other roots are given.