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 tho 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”.
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 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 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 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 = 3m = [[ 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 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 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"; elseecho"NO"; // This code is contributed by SanjuTomar ?> |
Output :
YES
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