Diagonally Dominant Matrix
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> |
YES
Time Complexity: O(N2)
Auxiliary Space: O(1), since no extra space has been taken.
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