# Is a square matrix strictly diagonally dominant?

A square matrix A is strictly diagonally dominant if for all rows the absolute value of the diagonal element in a row is strictly greater than than the sum of absolute value of the rest of the elements in that row.

In this posting, I show a MATLAB program that finds whether a square matrix is strictly diagonally dominant by using two different methods. These are academic ways to reinforce programming skills in a student.

The MATLAB program can be downloaded as a Mfile (better to download it, as single quotes from the web-post do not translate correctly with the MATLAB editor). The html file showing the mfile and the command window output is also available.

%% IS A GIVEN SQUARE MATRIX STRICTLY DIAGONALLY DOMINANT?
% Language : Matlab 2007a
% Authors : Autar Kaw
% Last Revised : November 25, 2008
% Abstract: This program shows you two ways of finding out
% if a square matrix is diagonally dominant. A square matrix is
% diagonally dominant if for all rows the absolute value of the
% diagonal element in a row is strictly greater than than the sum
% of absolute value of the rest of the elements in that row
clc
clear all
disp(‘This program shows you two ways of finding out’)
disp(‘if a square matrix is diagonally dominant. A square matrix is’)
disp(‘diagonally dominant if for all rows the absolute value of the’)
disp(‘diagonal element in a row is strictly greater than than the sum’)
disp(‘of absolute value of the rest of the elements in that row’)
disp(‘ ‘)
%% INPUTS
% The square matrix
A=[-12 1 -7 2;1 3.4 1.1 1.1; 1 0 -4.5 0;10 1 1 10];
disp (‘INPUTS’)
disp(‘Here is the square matrix’)
A
disp(‘ ‘)

%% FIRST SOLUTION
% This is based on finding for how many rows the condition
% the absolute value of the diagonal element in a row is
% strictly greater than than the sum of absolute value
% of the rest of the elements in that row.

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);
% count = for how many rows is the inequality met that
% the absolute value of the diagonal element in a row is
% strictly greater than than the sum of absolute value
% of the rest of the elements in that row
count=0;
for i=1:1:n
sumrow=0;
for j=1:1:n
if i~=j
sumrow=sumrow+abs(A(i,j));
end
end
if abs(A(i,i))>sumrow
count=count+1;
end
end
disp(‘FIRST WAY’)
if count==n
disp(‘Matrix is strictly diagonal dominant’)
else
disp(‘Matrix is NOT strictly diagonal dominant’)
end

%% SECOND SOLUTION
% This is based on finding for if for any row the condition
% the absolute value of the diagonal element in a row is
% strictly greater than than the sum of absolute value
% of the rest of the elements in that row is NOT met

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);
% flag = keeps track if the condition is not met
% flag = 1 if matrix is strictly diagonally dominant
% flag = 2 if matrix is not strictly diagonally dominant

% Assuming matrix is strictly diagonally dominant
flag=1;
for i=1:1:n
sumrow=0;
for j=1:1:n
if i~=j
sumrow=sumrow+abs(A(i,j));
end
end
% As soon as the condition is not met, it is not a strictly
% diagonally dominant matrix
if abs(A(i,i))<=sumrow
flag=2;
break;
end
end
disp(‘ ‘)
disp(‘SECOND WAY’)
if flag==1
disp(‘Matrix is strictly diagonal dominant’)
else
disp(‘Matrix is NOT strictly diagonal dominant’)
end

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu and the textbook on Numerical Methods with Applications available from the lulu storefront.

## Author: Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.

## 4 thoughts on “Is a square matrix strictly diagonally dominant?”

1. john says:

The code works just fine, but how can I change a matrix to a diagonally dominant one using matlab. Any available codes ?

Regards

_

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1. I do not have code for that. But remember, that every square matrix cannot be made diagonally dominant just by exchanging rows.

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2. Baffoe says:

Rs every 2 by 2 matrice always symmetric

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