Analytic Results for the Eigenvalues of Certain Tridiagonal Matrices

@article{Willms2008AnalyticRF,
  title={Analytic Results for the Eigenvalues of Certain Tridiagonal Matrices},
  author={Allan R. Willms},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2008},
  volume={30},
  pages={639-656}
}
The eigenvalue problem for a certain tridiagonal matrix with complex coefficients is considered. The eigenvalues and eigenvectors are shown to be expressible in terms of solutions of a certain scalar trigonometric equation. Explicit solutions of this equation are obtained for several special cases, and further analysis of this equation in several other cases provides information about the distribution of eigenvalues. 

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References

SHOWING 1-10 OF 10 REFERENCES

On the eigenvalues of some tridiagonal matrices

EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES

The eigenvalues and the corresponding eigenvectors of several tridiagonal matrices are derived by the method of symbolic calculus in (1) by solving the inequality of the following type: For α ≥ 1, β ≥ 1 using LaSalle's inequality.

The Characteristic Polynomial of Some Perturbed Tridiagonal k-Toeplitz Matrices 1

We generalize some recent results on the spectra of tridiagonal matrices, providing explicit expressions for the characteristic polynomial of some perturbed tridiagonal k-Toeplitz matrices. The

EIGENVALUES AND EIGENVECTORS OF TRIDIAGONAL MATRICES

This paper is continuation of previous work by the present author, where explicit formulas for the eigenvalues associated with several tridiagonal matrices were given. In this paper the associated

Matrix Analysis and Applied Linear Algebra

The author presents Perron-Frobenius theory of nonnegative matrices Index, a theory of matrices that combines linear equations, vector spaces, and matrix algebra with insights into eigenvalues and Eigenvectors.

Spectral theory for the differential equations of simple birth and death processes

  • W. LedermannG. Reuter
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1954
The enumerably infinite system of differential equations describing a temporally homogeneous birth and death process in a population is treated as the limiting case of one or the other of two finite

Numerical Analysis

This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis, and describes the accuracy, efficiency and robustness of these algorithms.

Partial Difference Equations.

Numerical Analysis

The voltage sensor in voltage-dependent ion channels.

The theoretical basis of the energy coupling between the electric field and the voltage is presented, which allows the interpretation of the gating charge that moves in one channel, and the novel results on lanthanide-based resonance energy transfer that show small distance changes between residues in the channel molecule.