Structured low rank updates of tridiagonal Toeplitz matrices

Some special low rank updates of tridiagonal Toeplitz matrices are considered that occur symmetrically on the first row and first column or the last row and the last column. Emphasis is given to the case of symmetric matrices. The fact that these matrices have known spectral data is used to provide exact formulas for eigenvalue approximation expressions when the updates are of sufficiently small magnitude. Finally, for some special forms of these updates, a complete characterization of eigenvalues and eigenvectors is given.