Exact eigenvalue and eigenvector solutions of certain tridiagonal generalized eigenvalue problems

It is known that certain tridiagonal matrices have exact eigenvalues and eigenvectors. There are sixteen documented tridiagonal matrix families, from the discretization of the one-dimensional Helmholtz equation that possess such properties. Extended members of these matrices share a same set of eigenvectors making them commutative with respective to matrix multiplication. We may therefore construct, in a fairly straightforward way, exact closed-form solutions of certain tridiagonal generalized matrix eigenvalue problems.