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”.
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"; elseecho"NO";// This code is contributed by SanjuTomar?> |
Output :
YES
Time Complexity: O(N2)
Auxiliary Space: O(1)
Please refer complete article on Diagonally Dominant Matrix for more details!
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