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: 0.94). 01/2008; 429(8-9):2221-2226. DOI: 10.1016/j.laa.2008.06.015
Source: arXiv


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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    • "which follows from the equality of dimensions of the Hilbert spaces above. This is a known identity (see for instance [12]), and we will present a proof in Appendix A, for completeness. 1 Moreover, equivalence of the operator algebras of the systems associated with (i) and (ii) allows us to make a stronger statement. "
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    ABSTRACT: We develop an equivalence between two Hilbert spaces: (i) the space of states of $U(1)^n$ Chern-Simons theory with a certain class of tridiagonal matrix of coupling constants (with corners) on $T^2$; and (ii) the space of ground states of strings on an associated mapping torus with $T^2$ fiber. The equivalence is deduced by studying the space of ground states of $SL(2,Z)$-twisted circle compactifications of $U(1)$ gauge theory, connected with a Janus configuration, and further compactified on $T^2$. The equality of dimensions of the two Hilbert spaces (i) and (ii) is equivalent to a known identity on determinants of tridiagonal matrices with corners. The equivalence of operator algebras acting on the two Hilbert spaces follows from a relation between the Smith normal form of the Chern-Simons coupling constant matrix and the isometry group of the mapping torus, as well as the torsion part of its first homology group.
    Journal of High Energy Physics 03/2014; 2014(7). DOI:10.1007/JHEP07(2014)010 · 6.11 Impact Factor
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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] "
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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.
    The electronic journal of linear algebra ELA 11/2012; 25(1):101-117. DOI:10.13001/1081-3810.1600 · 0.42 Impact Factor
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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. "
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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.
    Journal of Physics A Mathematical and Theoretical 10/2012; 46(25). DOI:10.1088/1751-8113/46/25/254004 · 1.58 Impact Factor
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