On the properties of matrices defining some classes of BVMs

It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property