Functions of Difference Matrices Are Toeplitz Plus Hankel

Abstract

When the heat equation and wave equation are approximated by $\bm{u}_t = -\bm{K} \bm{u}$ and $\bm{u}_{tt} = -\bm{K} \bm{u}$ (discrete in space), the solution operators involve $e^{-\bm{K}t}$, $\sqrt{\bm{K}}$, $\cos(\sqrt{\bm{K}}t)$, and $\mathrm{sinc}(\sqrt{\bm{K}}t)$. We compute these four matrices and find accurate approximations with a variety of boundary conditions. The second difference matrix $\bm{K}$ is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why $e^{\bm{-Kt}}$ also has a Hankel (anti-shift-invariant) part. Any symmetric choice of the four corner entries of $\bm{K}$ leads to Toeplitz plus Hankel in all functions $f(\bm{K})$. Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.