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A note on the eigenvalues of a special class of matrices

Cuminato, J.A. and McKee, S. (2010) A note on the eigenvalues of a special class of matrices. Journal of Computational and Applied Mathematics, 234 (9). pp. 2724-2731. ISSN 0377-0427

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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 @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.