A typical step in matrix algebra is elimination, and its description as a triangular factorization. For a doubly infinite banded Toeplitz matrix A, that step is made easy by factoring the polynomial a(z) whose coefficients come from the diagonals of A. What to do if A is not Toeplitz?

A nice case is a permutation matrix (on Z). Which is the main diagonal? For the (Toeplitz) example of a shift matrix, the main diagonal contains the 1's. We identify the correct diagonal for every banded permutation. Then we consider banded matrices (not Toeplitz!) as operators on L2(Z) and ask about their factorization.

A special case is when the inverse of A is also banded -- these matrices factor into block-diagonal matrices. The help coming from analysis is the theory of Fredholm operators.