This paper is about analytic properties of single transfer matrices
originating from general block-tridiagonal or banded matrices. Such matrices
occur in various applications in physics and numerical analysis. The
eigenvalues of the transfer matrix describe localization of eigenstates and are
linked to the spectrum of the block tridiagonal matrix by a determinantal
identity, If the block tridiagonal matrix is invertible, it is shown that half
of the singular values of the transfer matrix have a lower bound exponentially
large in the length of the chain, and the other half have an upper bound that
is exponentially small. This is a consequence of a theorem by Demko, Moss and
Smith on the decay of matrix elements of inverse of banded matrices.Comment: To appear in J. Phys. A: Math. and Theor. (Special issue on Lyapunov
Exponents, edited by F. Ginelli and M. Cencini). 16 page
