The Main Diagonal of a Permutation Matrix
  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. 
Marko Lindner, Gilbert Strang
math.NA