Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽

10.3390/sym13050870 ◽

2021 ◽

Vol 13 (5) ◽

pp. 870

Author(s):

Diego Caratelli ◽

Paolo Emilio Ricci

Keyword(s):

Functional Analysis ◽

Computer Algebra ◽

Toeplitz Matrices ◽

Test Cases ◽

Recursive Procedure ◽

Tridiagonal Matrices ◽

Special Cases ◽

The Matrix ◽

Matrix Eigenvalues ◽

Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

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A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

Numerical Algorithms ◽

10.1007/s11075-021-01130-9 ◽

2021 ◽

Author(s):

Sven-Erik Ekström ◽

Paris Vassalos

Keyword(s):

Distribution Function ◽

Asymptotic Distribution ◽

Numerical Experiments ◽

Toeplitz Matrices ◽

Future Research ◽

Research Directions ◽

Numerical Examples ◽

Working Hypothesis ◽

Wide Range ◽

Future Research Directions

AbstractIt is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

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Explicit and asymptotic formulas for LDMt factorization of banded toeplitz matrices

Linear Algebra and its Applications ◽

10.1016/0024-3795(95)94777-5 ◽

1995 ◽

Vol 222 ◽

pp. 97-126 ◽

Cited By ~ 3

Author(s):

James Lee Hafner

Keyword(s):

Toeplitz Matrices ◽

Asymptotic Formulas

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LIGHT COMPOSITE HIGGS: LCH @ LHC

International Journal of Modern Physics A ◽

10.1142/s0217751x05029162 ◽

2005 ◽

Vol 20 (27) ◽

pp. 6133-6148 ◽

Cited By ~ 18

Author(s):

FRANCESCO SANNINO

Keyword(s):

Gauge Group ◽

Higgs Mass ◽

Cold Dark Matter ◽

Precision Measurements ◽

Chiral Phase ◽

Symmetric Representation ◽

Composite Higgs ◽

Higher Dimensional ◽

The One ◽

Intrinsic Scale

Here I summarize some of the salient features of technicolor theories with technifermions in higher dimensional representations of the technicolor gauge group. The expected phase diagram as function of number of flavors and colors for the two index (anti)symmetric representation of the gauge group is reviewed. After having constructed the simplest walking technicolor theory one can show that it is not at odds with the precision measurements. The simplest theory also requires, for consistency, a fourth family of heavy leptons. The latter may result in an interesting signature at LHC. In the case of a fourth family of leptons with ordinary lepton hypercharge the new heavy neutrino can be a natural candidate of cold dark matter. New theories will also be proposed in which the critical number of flavors needed to enter the conformal window is higher than in the one with fermions in the two-index symmetric representation, but lower than in the walking technicolor theories with fermions only in the fundamental representation of the gauge group. Due to the near conformal/chiral phase transition the composite Higgs is very light compared to the intrinsic scale of the technicolor theory. For the two technicolor theory the composite Higgs mass is predicted not to exceed 150 GeV.

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