Transfer matrices and tridiagonal-block Hamiltonians with periodic and scattering boundary conditions

General Hamiltonian matrices with tridiagonal block structure and the associated transfer matrices are investigated in the cases of periodic and scattering boundary conditions. They arise from tight binding models with finite range hopping in one or more dimensions of space, in the presence of a Aharonov-Bohm flux or in multichannel scattering. An identity relating the characteristic equation of the periodic Hamiltonian with that of the transfer matrix is found, allowing a detailed analysis of the bands. A velocity matrix is defined, with properties relevant for the band structure, or for the channel structure in the scattering problem.