A note on the eigenvalues of a special class of matrices |
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Authors: | JA Cuminato |
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Institution: | a Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computaç ão, Universidade de São Paulo, São Carlos, Brazil b Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UK |
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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 σ, the stability parameter. As the order of the matrix is general, and the parameter σ 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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Keywords: | 65F15 |
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