Article

# Determinants of block tridiagonal matrices

Dipartimento di Fisica, Università degli Studi di Milano and INFN, Sezione di Milano, Via Celoria 16, Milano, Italy
(Impact Factor: 0.94). 01/2008; 429(8-9):2221-2226. 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).

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Available from: Luca Guido Molinari
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• "n = 2k : 44n − 126 44n − 258 for even k for odd k n = 2k + 1 44n − 159 44n − 291 for even k for odd k Now we describe Molinari's [6] "
##### Article: Determinants of multidiagonal matrices
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ABSTRACT: The formulas presented in [Molinari, L.G. Determinants of block tridiagonal matri-ces. Linear Algebra Appl., 2008; 429, 2221–2226] for evaluating the determinant of block tridiagonal matrices with (or without) corners are used to derive the determinant of any multidiagonal matri-ces with (or without) corners with some specified non-zero minors. Algorithms for calculation the determinant based on this method are given and properties of the determinants are studied. Some applications are presented.
Full-text · Article · Nov 2012 · The electronic journal of linear algebra ELA
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• "This relation among characteristic polynomials is a " duality identity " as it exchanges the roles of the parameters z and E among the two matrices: z is an eigenvalue of T (E) if and only if E is an eigenvalue of the block tridiagonal matrix H(z). I gave different proofs of it [7] [24] [25]. With z = 1 it is a tool for computing determinants of block tridiagonal or banded matrices with corners. "
##### Article: Identities and exponential bounds for transfer matrices
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ABSTRACT: This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity, If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of inverse of banded matrices.
Full-text · Article · Oct 2012 · Journal of Physics A Mathematical and Theoretical
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##### Article: Large system asymptotics of persistent currents in mesoscopic quantum rings
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ABSTRACT: We consider a one-dimensional mesoscopic quantum ring filled with spinless electrons and threaded by a magnetic flux, which carries a persistent current at zero temperature. The interplay of Coulomb interactions and a single on-site impurity yields a non-trivial dependence of the persistent current on the size of the ring. We determine numerically the asymptotic power law for systems up to 32000 sites for various impurity strengths and compare with predictions from Bethe Ansatz solutions combined with Bosonization. The numerical results are obtained using an improved functional renormalization group (fRG) method. We apply the density matrix renormalization group (DMRG) and exact diagonalization methods to benchmark the fRG calculations. We use DMRG to study the persistent current at low electron concentrations in order to extend the validity of our results to quasi-continuous systems. We briefly comment on the quality of calculated fRG ground state energies by comparison with exact DMRG data. Comment: REVTex, 12 pages, 12 figs, accepted in Phys. Rev. B
Full-text · Article · Feb 2009 · Physical review. B, Condensed matter