Before revision: We present analytical solutions to two classes of generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices.
After revision: It is well-known that the finite difference discretization of the Laplacian eigenvalue problem -\Delta u = \lambda u leads to a matrix eigenvalue problem (EVP) A x= \lambda x where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \mbox{%DIFAUXCMD strang2014functions analytical solutions to two classes of generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices.