A note on the eigenvalues of a special class of matrices
Given \(r>0\), let \(A=A(r)\) and \(B=B(r)\) be the \(m\times m\) tridiagonal matrices with main diagonal \((1\pm\frac{3}{2}r,1\pm r,\dots,1\pm r,1\pm\frac{3}{2}r)\) and each remaining nonzero entry \(\mp\frac{1}{2}r\). The upper signs refer to \(A\) and the lower to \(B\). The authors prove that the eigenvalues of \(M(r)=A^{-1}B\) are real and less than 1 of absolute value. This has earlier been proved by the authors jointly with \textit{C. M. Oishi} and \textit{J. Y. Yuan} [Numer. Linear Algebra Appl. 15, No.~10, 945--967 (2008; Zbl 1212.35184)]. They also construct such a nonsymmetric full \(m\times m\) matrix \(N(\sigma)\), \(\sigma=r/(2-r)\), \(0<r<2\), that is similar to \(M(r)\). So also the eigenvalues of \(N(\sigma)\) are real and less than 1 of absolute value. The motivation to study eigenvalues of these matrices arises from analyzing stability of a variant of the Crank-Nicolson method for the heat equation on a staggered grid.
scientific article; zbMATH DE number 5724542