The inverse of banded matrices

Abstract

The inverses of r-banded matrices, for r = 1, 2, 3 have been thoroughly investigated as one can see from the references we provide. Let B-r,B- n(1 <= r <= n) be an n x n matrix of entries {a(j)(i)}, -r <= I <= r, 1 <= j <= r, with the remaining un-indexed entries all zeros. In this paper, generalizing a method of Mallik (1999) [5], we give the LU factorization and the inverse of the matrix B-r,B- n (if it exists). Our results are valid for an arbitrary square matrix (taking r = 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 B-r,B-n. (c) 2012 Elsevier B.V. All rights reserved.