A note on the eigenvalues of a special class of matrices

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1,1] for all values of m (the order of the matrix) and all values of a positive parameter @s, the stability parameter. As the order of the matrix is general, and the parameter @s lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.
Original languageEnglish
Pages (from-to)2724-2731
Number of pages7
JournalJournal of Computational and Applied Mathematics
Volume234
Issue number9
DOIs
Publication statusPublished - Sep 2010

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Eigenvalue
Crank-Nicolson Method
Nonsymmetric Matrix
Staggered Grid
Test Set
Unit circle
Real Line
Heat Equation
Class
Hot Temperature

Keywords

  • 65F15
  • Crank-Nicolson
  • Eigenvalues
  • Special matrices
  • Tridiagonal matrices

Cite this

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title = "A note on the eigenvalues of a special class of matrices",
abstract = "In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1,1] for all values of m (the order of the matrix) and all values of a positive parameter @s, the stability parameter. As the order of the matrix is general, and the parameter @s lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.",
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author = "J.A. Cuminato and S. McKee",
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A note on the eigenvalues of a special class of matrices. / Cuminato, J.A.; McKee, S.

In: Journal of Computational and Applied Mathematics, Vol. 234, No. 9, 09.2010, p. 2724-2731.

Research output: Contribution to journalArticle

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AB - In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1,1] for all values of m (the order of the matrix) and all values of a positive parameter @s, the stability parameter. As the order of the matrix is general, and the parameter @s lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.

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KW - Special matrices

KW - Tridiagonal matrices

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