  By counting 1's in the "right half" of $2w$ consecutive rows, we locate the
main diagonal of any doubly infinite permutation matrix with bandwidth $w$.
Then the matrix can be correctly centered and factored into block-diagonal
permutation matrices. Part II of the paper discusses the same questions for the
much larger class of band-dominated matrices. The main diagonal is determined
by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is
determined "at infinity" in general, but from only $2w$ rows for banded
permutations.
