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.**

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john

said: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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Autar Kaw

said: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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Baffoe

said:Rs every 2 by 2 matrice always symmetric

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Autar Kaw

said:No, a matrix like [2 3; 5 6] is not symmetric

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