# Exact diagonalization of tridiagonal centrosymmetric matrices

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.

• Lanczos is a method that takes a symmetric matrix and construct a tridiagonal matrix similar to it (or, more often, a smaller tridiagonal matrix which is "approximately similar" to it, if one terminates it earlier). What you ask for, instead, is a method that takes a tridiagonal matrix and diagonalizes it. Oct 15, 2015 at 21:32
• By the way - are your matrices symmetric? Oct 15, 2015 at 21:32
• Yes, and not only the matrix is symmetric, but symmetric about its center (which is centrosymmetry=cs). In fact, for a generic cs-$N\times N$ matrix, I only have $N(N-1)/4$ independent matrix elements. Oct 16, 2015 at 5:10