A Sufficient Condition for Nonvanishing of Determinants

Abstract

In this note we derive sufficient conditions for a diagonally dominant reducible matrix to be nonsingular. 1. Throughout this note we are concerned with A = (aij), an «xn matrix which is diagonally dominant and where (1.1) J = lieN \aH\> J \atA^0 where 7V={1, 2, ■•■,«}. If J=N, A is strictly diagonally dominant and then the Gersgorin circle theorem implies that the determinant of A does not vanish [1, p. 106]. If A is irreducible, Taussky [5] has shown that A is nonsingular. In this note, we prove the following theorem. Theorem. Let the matrix A be such that for each i$J there is a sequence of nonzero elements of A of the form au , at., • • • , air¡ withjeJ. Then A is nonsingular. 2. We need the following lemma and results to prove the Theorem. Lemma 1. Let A satisfy the conditions of the theorem. Then for any nonempty subset L of N such that LC\J= 0, there is a nonzero element au with z" e L andj $ L. Proof. Let L be a nonempty subset of N such that L nJ= 0. Choose /, e L, then i1 $ J and, hence, there is a sequence of nonzero elements of A of the form aiiH, aÍ2Í3, •••,«