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