Exact eigensystems for some matrices arising from discretizations

It is known, for example, that the eigenvalues of the N×N matrix A, arising in the discretization of the wave equation, whose only nonzero entries are Akk+1=Ak+1k=-1,k=1,…,N-1, and Akk=2,k=1,…,N, are 2{1-cos[pπ/(N+1)]} with corresponding eigenvectors v(p) given by . We show by considering a simple finite difference approximation to the second derivative and using the summation formulae for sines and cosines that these and other similar formulae arise in a simple and unified way.