7,779 research outputs found
The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices
Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations
Total positivity and spectral theory for Toeplitz Hessenberg matrix ensembles
In this paper we define and lay the groundwork for studying a novel matrix
ensemble: totally positive Hessenberg Toeplitz operators, denoted TPHT. This is
the intersection of two ensembles that have been significantly explored:
totally positive Hessenberg matrices (TPH) and Hessenberg Toeplitz matrices
(HT). TPHT has a rich linear algebraic and spectral structure that we describe.
Along the way we find some previously unnoticed connections between certain
Toeplitz normal forms for matrices and Lie theoretic interpretations. We also
numerically study the spectral asymptotics of TPH matrices via the TPHT
ensemble and use this to open a study of TPHT with random symbols
Normal form for GLT sequences, functions of normal GLT sequences, and spectral distribution of perturbed normal matrices
The theory of generalized locally Toeplitz (GLT) sequences is a powerful
apparatus for computing the asymptotic spectral distribution of matrices
arising from numerical discretizations of differential equations. Indeed, when
the mesh fineness parameter tends to infinity, these matrices give
rise to a sequence , which often turns out to be a GLT sequence. In
this paper, we extend the theory of GLT sequences in several directions: we
show that every GLT sequence enjoys a normal form, we identify the spectral
symbol of every GLT sequence formed by normal matrices, and we prove that, for
every GLT sequence formed by normal matrices and every continuous
function , the sequence is again a GLT
sequence whose spectral symbol is , where is the spectral
symbol of . In addition, using the theory of GLT sequences, we prove
a spectral distribution result for perturbed normal matrices
A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth
This paper can be thought of as a remark of \cite{llw}, where the authors
studied the eigenvalue distribution of random block Toeplitz band
matrices with given block order . In this note we will give explicit density
functions of when the bandwidth grows
slowly. In fact, these densities are exactly the normalized one-point
correlation functions of Gaussian unitary ensemble (GUE for short).
The series can be seen
as a transition from the standard normal distribution to semicircle
distribution. We also show a similar relationship between GOE and block
Toeplitz band matrices with symmetric blocks.Comment: 6 page
A Simple Proof of the Classification of Normal Toeplitz Matrices
We give an easy proof to show that every complex normal Toeplitz matrix is
classified as either of type I or of type II. Instead of difference equations
on elements in the matrix used in past studies, polynomial equations with
coefficients of elements are used. In a similar fashion, we show that a real
normal Toeplitz matrix must be one of four types: symmetric, skew-symmetric,
circulant, or skew-circulant. Here we use trigonometric polynomials in the
complex case and algebraic polynomials in the real case.Comment: 5 page
Toeplitz separability, entanglement, and complete positivity using operator system duality
A new proof is presented of a theorem of L.~Gurvits, which states that the
cone of positive block-Toeplitz matrices with matrix entries has no entangled
elements. The proof of the Gurvits separation theorem is achieved by making use
of the structure of the operator system dual of the operator system
of Toeplitz matrices over the complex field, and by
determining precisely the structure of the generators of the extremal rays of
the positive cones of the operator systems C(S^1)^{(n)}\omin\B(\H) and
C(S^1)_{(n)}\omin\B(\H), where \H is an arbitrary Hilbert space and
is the operator system dual of . Our approach also
has the advantage of providing some new information concerning positive
Toeplitz matrices whose entries are from \B(\H) when \H has infinite
dimension. In particular, we prove that normal positive linear maps on
\B(\H) are partially completely positive in the sense that is
positive whenever is a positive Toeplitz matrix with entries
from \B(\H). We also establish a certain factorisation theorem for positive
Toeplitz matrices (of operators), showing an equivalence between the Gurvits
approach to separation and an earlier approach of T.~Ando to universality
- …