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     

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.     

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.     

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 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.     

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.     

Bounds for eigenvalues of symmetric block Jacobi scaled matrices

The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles. Translated by L. Yu. Kolotilina. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 18–25.     

On close eigenvalues of tridiagonal matrices

Summary. A symmetric tridiagonal matrix with a multiple eigenvalue must have a zero subdiagonal element and must be a direct sum of two complementary blocks, both of which have the eigenvalue. Yet it is well known that a small spectral gap does not necessarily imply that some is small, as is demonstrated by the Wilkinson matrix. In this note, it is shown that a pair of close eigenvalues can only arise from two complementary blocks on the diagonal, in spite of the fact that the coupling the two blocks may not be small. In particular, some explanatory bounds are derived and a connection to the Lanczos algorithm is observed. The nonsymmetric problem is also included. Received April 8, 1992 / Revised version received September 21, 1994     

On the eigenvalue problem for toeplitz matrices generated by rational functions

Formulas are given for the characteristic polynomials {p_n (λ)}and the eigenvectors of the family {T_n }of Toeplitz matrices generated by a formal Laurent series of rational function R(z). The formulas are in terms of the zeros of a certain fixed polynomial with coefficients which are simple functions of λ and the coefficients of R(z). The complexity of the formulas is independent ofn.