From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: help - diagonally dominant matrix is nonsingular
Date: 27 Jan 2000 01:40:26 -0500
Newsgroups: sci.math.num-analysis
Summary: Banach Lemma: matrices near the identity are invertible

In article <388FCE55.DACD596D@cae.ca>, dans  <dans@cae.ca> wrote:
:Hi,
:
:  Can someone help me with a solution?
:I have a matrix A that is diagonally dominant : SUM(j=1,N; j!=i)
:|A[i][j]| < |A[i][i]|.
:I will like to prove that the matrix A is nonsingular by using the
:Banach Lemma.
:
:Thanks,
:Dan Sirbu
:
:Banach Lemma : ||B|| < 1 => B is nonsingular.

Correction: ||B|| < 1 => (I-B) is nonsingular.

Define D = diag(A), that is, 

   D[i][i]=A[i][i] and for i different from j, D[i][j]=0.

D is nonsingular, and you consider B = I - D^(-1) * A, with the matrix
norm subordinate to the maximum norm of vectors (i.e. maximum of sums of
absolute values within rows).

Good luck, ZVK(Slavek).