Article

Determinants of block tridiagonal matrices

Dipartimento di Fisica, Università degli Studi di Milano and INFN, Sezione di Milano, Via Celoria 16, Milano, Italy
Linear Algebra and its Applications (Impact Factor: 0.98). 01/2008; DOI: 10.1016/j.laa.2008.06.015
Source: arXiv

ABSTRACT An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).

1 Follower
 · 
476 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Recent analytical and numerical work have shown that the spectrum of the random non-hermitian Hamiltonian on a ring which models the physics of vortex line pinning in superconductors is one dimensional. In the maximally non-hermitian limit, we give a simple “one-line” proof of this feature. We then study the spectral curves for various distributions of the random site energies. We find that a critical transition occurs when the average of the logarithm of the random site energy squared vanishes. For a large class of probability distributions of the site energies, we find that as the randomness increases the energy at which the localization-delocalization transition occurs increases, reaches a maximum, and then decreases. The Cauchy distribution studied previously in the literature does not have this generic behavior. We determine γc1, the critical value of the randomness at which “wings” first appear in the energy spectrum. For distributions, such as Cauchy, with infinitely long tails, we show that γc1 = 0+. We determine the density of eigenvalues on the wings for any probability distribution. We show that the localization length on the wings diverge generically as as E approaches . These results are all obtained in the maximally non-hermitian limit but for a generic class of probability distributions of the random site energies.
    Nuclear Physics B 07/1999; 552(3-552):599-623. DOI:10.1016/S0550-3213(99)00246-1 · 3.95 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this article the author shows that under certain conditions a three-term recurrence for a tridiagonal matrix becomes a two-term recurrence. Using this new recurrence, the possibility of the LU factorization of any tridiagonal matrix is now easy to investigate. The positive definiteness of any real symmetric tridiagonal matrix is now easy to check. An algorithm for solving any linear system with positive definite tridiagonal matrix is given. Some numerical examples are given.
    Applied Mathematics and Computation 07/2003; 139(2):503-511. DOI:10.1016/S0096-3003(02)00212-6 · 1.60 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: We use a transfer-matrix method to study defects in a tight-binding model of carbon nanotubes. We calculate the reflection coefficient R for a simple barrier created by a pointlike defect of strength E in armchair (Na,Na) and zigzag (Na,0) nanotubes for the whole range of energy ω and arbitrary number of conducting channels. We find that R scales at the Fermi level (i.e., ω=0) as R=s(E/t)2/Na2 (t being the hopping parameter), where s≈1/6 (for the armchair nanotubes) and s≈1/2 (for the zigzag nanotubes). We also perform a similar calculation for a “5-77-5” defect and find the results to be like the ones obtained for a strong point defect with E=6t.
    Physical Review B 01/1999; 59(4). DOI:10.1103/PhysRevB.59.3241 · 3.66 Impact Factor

Full-text (2 Sources)

Download
97 Downloads
Available from
May 22, 2014