A note on the eigenvalues of a special class of matrices

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.