On the eigenvalues of some tridiagonal matrices

Fonseca, Carlos M. da

doi:10.1016/j.cam.2005.08.047

Cited by 77 publications

(36 citation statements)

References 4 publications

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“…Meet matrices were defined by Rajarama Bhat [20] for the first time and in this same article MIN matrices are considered as an example. da Fonseca [7] studies the eigenvalues of certain MIN and MAX matrices via their matrix inverses, and in [9] bounds for the values of trigonometric functions are found by underestimating the smallest eigenvalue of a MIN matrix. Also the connection between generalized Fibonacci numbers and the characteristic polynomials of MIN and MAX matrices have been studied recently, see [2].…”

Section: Introductionmentioning

confidence: 99%

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Mattila

Haukkanen

2016

Special Matrices

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Let T = {z 1 , z 2 , . . . , zn} be a finite multiset of real numbers, where z 1 ≤ z 2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(z i , z j ) and max(z i , z j ) as their ij entries, respectively. We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether it would be possible to prove these same results by using elementary matrix methods only. In many cases the answer is positive.

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Section: Introductionmentioning

confidence: 99%

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Mattila

Haukkanen

2016

Special Matrices

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“…Both spectral expressions generalize many particular cases known in the literature, as for example it can be seen in [5,7,[9][10][11][12]14,[28][29][30][31][32][33].…”

Section: Preliminariesmentioning

confidence: 52%

A multi-temperature kinetic Ising model and the eigenvalues of some perturbed Jacobi matrices

Fonseca

Kouachi

Mazilu

et al. 2015

Applied Mathematics and Computation

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“…, N, (26) where the phase, defined in the interval φ 1 λ (k) ∈] − π 2 , π 2 ], represents the momentum shift in relation to the usual φ 1 λ (k) = 0 case, for which one recovers k n = nπ N +1 . For every band λ we solve (26) for each n to find the set of allowed k n values within the Reduced Brillouin Zone (RBZ), k n ∈ [0, π[. An example of the geometrical determination of the k states, for a system with N = 10 unit cells and δ = π 2 , is shown in Fig.…”

Section: Ssh4 Modelmentioning

confidence: 99%

Analytical solution of open crystalline linear 1D tight-binding models

Marques

Dias

2020

J. Phys. A: Math. Theor.

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A method for finding the exact analytical solutions for the bulk and edge energy levels and corresponding eigenstates for all commensurate Aubry-André/Harper single-particle models under open boundary conditions is presented here, both for integer and non-integer number of unit cells. The solutions are ultimately found to be dependent on the behavior of phase factors whose compact formulas, provided here, make this method simple to implement computationally. The derivation employs the properties of the Hamiltonians of these models, all of which can be written as Hermitian block-tridiagonal Toeplitz matrices. The concept of energy spectrum is generalized to incorporate both bulk and edge bands, where the latter are a function of a complex momentum. The method is then extended to solve the case where one of these chains is coupled at one end to an arbitrary cluster/impurity. Future developments based on these results are discussed.

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On the eigenvalues of some tridiagonal matrices

Cited by 77 publications

References 4 publications

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

A multi-temperature kinetic Ising model and the eigenvalues of some perturbed Jacobi matrices

Analytical solution of open crystalline linear 1D tight-binding models

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