Criteria for invertibility of diagonally dominant matrices

Abstract

We develop some basic properties of finite diagonally dominant matrices. These properties are used to establish a necessary and sufficient condition for a finite diagonally dominant matrix with nonzero diagonal entries to be singular. This condition relates the nonstrict diagonally dominant rows of the matrix to the difference between the principal arguments of the nonzero entries along each column in these rows. The infinite dimensional case is also studied, where a sufficient condition for the invertibility of the matrix operator in the sequence space c 0 defined by a diagonally dominant infinite matrix A with nonzero diagonal entries is introduced. This sufficient condition improves some of the earlier results. When the sequence space is l p , p ∈ [1, ∞), we establish a necessary and sufficient condition for the matrix operator in l p defined by the matrix A to have a bounded inverse on l p .