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Improving formulas for the eigenvalues of finite block-Toeplitz tridiagonal matrices

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Abstract

After a short overview, improvements (based on the Kronecker product) are proposed for the eigenvalues of (N × N) block-Toeplitz tridiagonal (block-TT) matrices with (K × K) matrix-entries, common in applications. Some extensions of the spectral properties of the Toeplitz-tridiagonal matrices are pointed-out. The eigenvalues of diagonalizable symmetric and skew-symmetric block-TT matrices are studied. Besides, if certain matrix square-root is well-defined, it is proved that every block-TT matrix with commuting matrix-entries is isospectral to a related symmetric block-TT one. Further insight about the eigenvalues of hierarchical Hermitian block-TT matrices, of use in the solution of PDEs, is also achieved.

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... where I M y −1 is the identity matrix and J M y −1 = tridiag(1, 0, 1). The eigenvalues of A are obtained from the following relation [36] ...
... For the sake of simplicity, let M x = M y := M and h x = h y := h; we can change Equations (36) and (37) as ...
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... Fig. 1b). In such a case, the Hamiltonian belongs to the symmetric block-Toeplitz tridiagonal matrices which can be displayed aŝ The eigenvalues of the above matrixĤ ex are [32] λ α n = 2J cos((n + 1)π/N + 1) + 2L cos(απ/4), ...
... . Then the eigenvalues will be [32] λ α n = 2J cos ((n + 1)π/N + 1) ...
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... The eigenvalues of the above matrixĤ ex are [32] λ α n = 2J cos((n + 1)π/N + 1) + 2L cos(απ/4), (34) where α = 1, 2, 3 and n = 0, 1, ..., N − 1. ...
... and C 1 0 = C 0 1 = JÎ 3×3 . Then the eigenvalues will be [32] λ α n = 2J cos ((n + 1)π/N + 1) + 2L cos (2πα/3), ...
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... For a larger system i.e. 50x100 sites, the matrix becomes 5000 x 5000 for which the diagonalization is computationally expensive for different values of J 2 . To diagonalize this large sparse matrix efficiently, we used the method given by Ref. [41]. The energy spectrum for the larger systemas shown in Fig. 2 (c-d), (g-h) -doesn't show any deviation apart from very small bulk oscillations in the region where αω > |J −J|. ...
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... We have studied spectral properties Toeplitz-tridiagonal matrices and eigenvalues of symmetric diagonalizable matrices [34]. ...
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Contenido: Introducción; Resolución de ecuaciones lineales; Problemas menos lineales de cuadrados; Problemas de valor propio no simétricos; El problema de valor propio simétrico y el valor singular de descomposición; Métodos iterativos para sistemas lineales; Métodos iterativos para problemas de valor propio.
  • A J Laub
A.J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, Philadelphia, USA, 2005.
  • C D Meyer
C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, USA, 20 0 0.