Three theorems dealing with transfer matrices in statistical mechanical systems are proved. The theorems state that the nonzero eigenvalues of transfer matrices formed through various prescriptions are identical. Hence it is possible to ascribe a physical meaning to all the eigenvalues of a transfer matrix, not just to the few largest eigenvalues. The first theorem states that the transfer matrix formed by building a system M layers at a time has as its only nonzero eigenvalues the eigenvalues of the transfer matrix formed by building the M layers of the system one at a time. This theorem relates the product of two nM×nM M-layer transfer matrices to the product of M one-layer M×M transfer matrices. The second theorem states that one of the nM×nM M-layer transfer matrices (for M>1) has only one nonzero eigenvalue. A procedure for finding this eigenvalue and all eigenvectors is given. The third theorem generalizes the first to the case where the chosen layering is not an integer multiple of the interaction length.
A novel numerical transfer matrix approach to obtain the critical behavior of Ising models in dimensions higher than two is described. The approach augments the physical transfer matrices with other matrices which do not necessarily have a straightforward physical interpretation. These additional matrices have approximately the same form as the physical transfer matrices, and for high dimensions may provide the only way of performing finite size scaling using transfer matrix calculations. Results are presented for the Ising model in three, four, and five dimensions.