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Diagonally Dominant Matrix

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  • Difficulty Level : Easy
  • Last Updated : 19 Aug, 2022
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In mathematics, a square matrix is said to be diagonally dominant if for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if 
 

For example, The matrix 
 

is diagonally dominant because 
|a11| ≥ |a12| + |a13| since |+3| ≥ |-2| + |+1| 
|a22| ≥ |a21| + |a23| since |-3| ≥ |+1| + |+2| 
|a33| ≥ |a31| + |a32| since |+4| ≥ |-1| + |+2|
Given a matrix A of n rows and n columns. The task is to check whether matrix A is diagonally dominant or not.

Examples : 

Input : A = { { 3, -2, 1 },
              { 1, -3, 2 },
              { -1, 2, 4 } };
Output : YES
Given matrix is diagonally dominant
because absolute value of every diagonal
element is more than sum of absolute values
of corresponding row.

Input : A = { { -2, 2, 1 },
              { 1, 3, 2 },
              { 1, -2, 0 } };
Output : NO

The idea is to run a loop from i = 0 to n-1 for the number of rows and for each row, run a loop j = 0 to n-1 find the sum of non-diagonal element i.e i != j. And check if diagonal element is greater than or equal to sum. If for any row, it is false, then return false or print “No”. Else print “YES”. 

Implementation:

C++




// CPP Program to check whether given matrix
// is Diagonally Dominant Matrix.
#include <bits/stdc++.h>
#define N 3
using namespace std;
 
// check the given  matrix is Diagonally
// Dominant Matrix or not.
bool isDDM(int m[N][N], int n)
{
    // for each row
    for (int i = 0; i < n; i++)
   {       
 
        // for each column, finding sum of each row.
        int sum = 0;
        for (int j = 0; j < n; j++)            
            sum += abs(m[i][j]);       
 
        // removing the diagonal element.
        sum -= abs(m[i][i]);
 
        // checking if diagonal element is less
        // than sum of non-diagonal element.
        if (abs(m[i][i]) < sum)
            return false;
        
    }
 
    return true;
}
 
// Driven Program
int main()
{
    int n = 3;
    int m[N][N] = { { 3, -2, 1 },
                    { 1, -3, 2 },
                    { -1, 2, 4 } };
 
    (isDDM(m, n)) ? (cout << "YES") : (cout << "NO");
 
    return 0;
}

Java




// JAVA Program to check whether given matrix
// is Diagonally Dominant Matrix.
import java.util.*;
 
class GFG {
     
    // check the given  matrix is Diagonally
    // Dominant Matrix or not.
    static boolean isDDM(int m[][], int n)
    {
        // for each row
        for (int i = 0; i < n; i++)
        {       
      
            // for each column, finding
            //sum of each row.
            int sum = 0;
            for (int j = 0; j < n; j++)            
                sum += Math.abs(m[i][j]);       
      
            // removing the diagonal element.
            sum -= Math.abs(m[i][i]);
      
            // checking if diagonal element is less
            // than sum of non-diagonal element.
            if (Math.abs(m[i][i]) < sum)
                return false;
        
        }
 
        return true;
    }
 
    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int n = 3;
        int m[][] = { { 3, -2, 1 },
                      { 1, -3, 2 },
                      { -1, 2, 4 } };
      
        if (isDDM(m, n))
             System.out.println("YES") ;
        else 
            System.out.println("NO");
     
    }
}
 
// This code is contributed by  Arnav Kr. Mandal.

Python3




# Python Program to check
# whether given matrix is
# Diagonally Dominant Matrix.
 
# check the given 
# matrix is Diagonally
# Dominant Matrix or not.
def isDDM(m, n) :
 
    # for each row
    for i in range(0, n) :        
     
        # for each column, finding
        # sum of each row.
        sum = 0
        for j in range(0, n) :
            sum = sum + abs(m[i][j])    
 
        # removing the
        # diagonal element.
        sum = sum - abs(m[i][i])
 
        # checking if diagonal
        # element is less than
        # sum of non-diagonal
        # element.
        if (abs(m[i][i]) < sum) :
            return False
 
    return True
 
# Driver Code
n = 3
m = [[ 3, -2, 1 ],
    [ 1, -3, 2 ],
    [ -1, 2, 4 ]]
 
if((isDDM(m, n))) :
    print ("YES")
else :
    print ("NO")
 
# This code is contributed by
# Manish Shaw(manishshaw1)

C#




// C# Program to check whether given matrix
// is Diagonally Dominant Matrix.
using System;
 
class GFG {
     
    // check the given matrix is Diagonally
    // Dominant Matrix or not.
    static bool isDDM(int [,]m, int n)
    {
        // for each row
        for (int i = 0; i < n; i++)
        {
     
            // for each column, finding
            //sum of each row.
            int sum = 0;
            for (int j = 0; j < n; j++)        
                sum += Math.Abs(m[i, j]);    
     
            // removing the diagonal element.
            sum -= Math.Abs(m[i, i]);
     
            // checking if diagonal element is less
            // than sum of non-diagonal element.
            if (Math.Abs(m[i,i]) < sum)
                return false;
         
        }
 
        return true;
    }
 
    // Driver program
    public static void Main()
    {
        int n = 3;
        int [,]m = { { 3, -2, 1 },
                    { 1, -3, 2 },
                    { -1, 2, 4 } };
     
        if (isDDM(m, n))
            Console.WriteLine("YES") ;
        else
            Console.WriteLine("NO");
     
    }
}
 
// This code is contributed by Vt_m.

PHP




<?php
// PHP Program to check whether
// given matrix is Diagonally
// Dominant Matrix.
 
// check the given  matrix
// is Diagonally Dominant Matrix or not.
function isDDM( $m, $n)
{
    // for each row
    for ($i = 0; $i < $n; $i++)
         
    {
        // for each column, finding
        // sum of each row.
        $sum = 0;
        for ( $j = 0; $j < $n; $j++)            
            $sum += abs($m[$i][$j]);    
 
        // removing the diagonal element.
        $sum -= abs($m[$i][$i]);
 
        // checking if diagonal element
        // is less than sum of non-diagonal
        // element.
        if (abs($m[$i][$i]) < $sum)
            return false;
    }
 
    return true;
}
 
// Driver Code
$n = 3;
$m = array(array( 3, -2, 1 ),
           array( 1, -3, 2 ),
           array( -1, 2, 4 ));
 
if((isDDM($m, $n)))
echo "YES";
else
echo"NO";
 
// This code is contributed by SanjuTomar
?>

Javascript




<script>
 
// JavaScript Program to check whether given matrix
// is Diagonally Dominant Matrix.
 
    // check the given matrix is Diagonally
    // Dominant Matrix or not.
    function isDDM(m, n)
    {
        // for each row
        for (let i = 0; i < n; i++)
        {       
        
            // for each column, finding
            //sum of each row.
            let sum = 0;
            for (let j = 0; j < n; j++)            
                sum += Math.abs(m[i][j]);       
        
            // removing the diagonal element.
            sum -= Math.abs(m[i][i]);
        
            // checking if diagonal element is less
            // than sum of non-diagonal element.
            if (Math.abs(m[i][i]) < sum)
                return false;
          
        }
   
        return true;
    }
 
// Driver code   
          
        let n = 3;
        let m = [[ 3, -2, 1 ],
                      [ 1, -3, 2 ],
                      [ -1, 2, 4 ]];
        
        if (isDDM(m, n))
             document.write("YES") ;
        else 
           document.write("NO");
             
</script>

Output

YES

 Time Complexity: O(N2)
Auxiliary Space: O(1), since no extra space has been taken.


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