It is said that one can diagonalize a tridiagonal matrix using the analytical Lanczos method http://arxiv.org/abs/cond-mat/9712283v1. In some references in it, they always say that the starting point is to choose a convenient initial state and from it one can construct the tridiagonal matrix.

The issue is the following: I want to diagonalize a tridiagonal matrix already set, which is also centrosymmetric. In this matrix in general I have $(N-1)/2$ independent parameters (where $N$ is the matrix dimension - square matrix). With these constraints, is it possible to diagonalize a generic matrix fulfilling these constraints?

References for academic purposes will be much appreciated.