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”.
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. |
Output :
YES
Time Complexity: O(N2)
Auxiliary Space: O(1)
Please refer complete article on Diagonally Dominant Matrix for more details!
