Abstract
We consider the inversion of block tridiagonal, block Toeplitz matrices and
comment on the behaviour of these inverses as one moves away from the
diagonal. Using matrix Möbius transformations, we first present an O(1)
representation (with respect to the number of block rows and block columns) for
the inverse matrix and subsequently use this representation to characterize the
inverse matrix. There are four symmetry-distinct cases where the blocks of the
inverse matrix (i) decay to zero on both sides of the diagonal, (ii) oscillate on
both sides, (iii) decay on one side and oscillate on the other and (iv) decay on
one side and grow on the other. This characterization exposes the necessary
conditions for the inverse matrix to be numerically banded and may also aid in
the design of preconditioners and fast algorithms. Finally, we present numerical
examples of these matrix types.