Exact eigensystems for some matrices arising from discretizations (Q999783)

Let \({n}\) be a positive integer. Let \(A\) be an \({n}\times{n}\) matrix whose only nonzero entries are \(a_{i(i+1)} = a_{(i+1)i} = {-1}\), \({i=1,2,\dots,n-1}\), and \(a_{ii} ={2}\), \({i=1,2,\dots,n}\). These matrices arising in the discretization of the wave equation. The eigenvalues of this \({n}\times{n}\) matrix are \(\lambda_{p} ={2}(1- \cos(\frac{{p}\pi}{n+1})\) with corresponding eigenvectors \(v^{p}_{i}= \sin(\frac{ip\pi}{n+1})\),\({i,p=1,2,\dots,n}\). The authors consider the eigenproblem for the operator \(-\frac{d^{2}}{dx^{2}}\) on the interval \([a,b]\), with several types of boundaries conditions at \({a}\) and \({b}\), where \(\frac{d^{2}u}{dx^{2}}+\lambda{u}={0}\). They use a simple finite difference approximation and using the summation formulae for sines and cosines that these formulae (for eigenvalues and eigenvectors) arise in a simple and unified way.