The inverse of banded matrices

The inverses of rr-banded matrices, for r=1,2,3r=1,2,3 have been thoroughly investigated as one can see from the references we provide. Let Br,nBr,n (1≤r≤n1≤r≤n) be an n×nn×n matrix of entries {aji}, −r≤i≤r−r≤i≤r, 1≤j≤r1≤j≤r, with the remaining un-indexed entries all zeros. In this paper, generalizing a method of Mallik (1999) [5], we give the LULU factorization and the inverse of the matrix Br,nBr,n (if it exists). Our results are valid for an arbitrary square matrix (taking r=nr=n), and so, we will give a new approach for computing the inverse of an invertible square matrix. Our method is based on Hessenberg submatrices associated to Br,nBr,n.