Given a 2D matrix, the task is to find Trace and Normal of matrix.
Normal of a matrix is defined as square root of sum of squares of matrix elements.
Trace of a n x n square matrix is sum of diagonal elements.
Examples :
Input : mat[][] = {{7, 8, 9}, {6, 1, 2}, {5, 4, 3}}; Output : Normal = 16 Trace = 11 Explanation : Normal = sqrt(7*7+ 8*8 + 9*9 + 6*6 + 1*1 + 2*2 + 5*5 + 4*4 + 3*3) = 16 Trace = 7+1+3 = 11 Input :mat[][] = {{1, 2, 3}, {6, 4, 5}, {2, 1, 3}}; Output : Normal = 10 Trace = 8 Explanation : Normal = sqrt(1*1 +2*2 + 3*3 + 6*6 + 4*4 + 5*5 + 2*2 + 1*1 + 3*3) Trace = 8(1+4+3)
C++
// C++ program to find trace and normal // of given matrix #include<bits/stdc++.h> using namespace std; // Size of given matrix const int MAX = 100; // Returns Normal of a matrix of size n x n int findNormal( int mat[][MAX], int n) { int sum = 0; for ( int i=0; i<n; i++) for ( int j=0; j<n; j++) sum += mat[i][j]*mat[i][j]; return sqrt (sum); } // Returns trace of a matrix of size n x n int findTrace( int mat[][MAX], int n) { int sum = 0; for ( int i=0; i<n; i++) sum += mat[i][i]; return sum; } // Driven source int main() { int mat[][MAX] = {{1, 1, 1, 1, 1}, {2, 2, 2, 2, 2}, {3, 3, 3, 3, 3}, {4, 4, 4, 4, 4}, {5, 5, 5, 5, 5}, }; cout << "Trace of Matrix = " << findTrace(mat, 5) << endl; cout << "Normal of Matrix = " << findNormal(mat, 5) << endl; return 0; } |
Java
// Java program to find trace and normal // of given matrix import java.io.*; class GFG { // Size of given matrix static int MAX = 100 ; // Returns Normal of a matrix of size n x n static int findNormal( int mat[][], int n) { int sum = 0 ; for ( int i= 0 ; i<n; i++) for ( int j= 0 ; j<n; j++) sum += mat[i][j]*mat[i][j]; return ( int )Math.sqrt(sum); } // Returns trace of a matrix of size n x n static int findTrace( int mat[][], int n) { int sum = 0 ; for ( int i= 0 ; i<n; i++) sum += mat[i][i]; return sum; } // Driven source public static void main (String[] args) { int mat[][] = {{ 1 , 1 , 1 , 1 , 1 }, { 2 , 2 , 2 , 2 , 2 }, { 3 , 3 , 3 , 3 , 3 }, { 4 , 4 , 4 , 4 , 4 }, { 5 , 5 , 5 , 5 , 5 }, }; System.out.println ( "Trace of Matrix = " + findTrace(mat, 5 )); System.out.println ( "Normal of Matrix = " + findNormal(mat, 5 )); } } // This code is contributed by vt_m. |
Python3
# Python3 program to find trace and # normal of given matrix import math # Size of given matrix MAX = 100 ; # Returns Normal of a matrix # of size n x n def findNormal(mat, n): sum = 0 ; for i in range (n): for j in range (n): sum + = mat[i][j] * mat[i][j]; return math.floor(math.sqrt( sum )); # Returns trace of a matrix of # size n x n def findTrace(mat, n): sum = 0 ; for i in range (n): sum + = mat[i][i]; return sum ; # Driver Code mat = [[ 1 , 1 , 1 , 1 , 1 ], [ 2 , 2 , 2 , 2 , 2 ], [ 3 , 3 , 3 , 3 , 3 ], [ 4 , 4 , 4 , 4 , 4 ], [ 5 , 5 , 5 , 5 , 5 ]]; print ( "Trace of Matrix =" , findTrace(mat, 5 )); print ( "Normal of Matrix =" , findNormal(mat, 5 )); # This code is contributed by mits |
C#
// C# program to find trace and normal // of given matrix using System; class GFG { // Returns Normal of a matrix of // size n x n static int findNormal( int [,]mat, int n) { int sum = 0; for ( int i = 0; i < n; i++) for ( int j = 0; j < n; j++) sum += mat[i,j] * mat[i,j]; return ( int )Math.Sqrt(sum); } // Returns trace of a matrix of size // n x n static int findTrace( int [,]mat, int n) { int sum = 0; for ( int i = 0; i < n; i++) sum += mat[i,i]; return sum; } // Driven source public static void Main () { int [,]mat = { {1, 1, 1, 1, 1}, {2, 2, 2, 2, 2}, {3, 3, 3, 3, 3}, {4, 4, 4, 4, 4}, {5, 5, 5, 5, 5}, }; Console.Write ( "Trace of Matrix = " + findTrace(mat, 5) + "\n" ); Console.Write( "Normal of Matrix = " + findNormal(mat, 5)); } } // This code is contributed by nitin mittal. |
PHP
<?php // PHP program to find trace and // normal of given matrix // Size of given matrix $MAX = 100; // Returns Normal of a // matrix of size n x n function findNormal( $mat , $n ) { $sum = 0; for ( $i = 0; $i < $n ; $i ++) for ( $j = 0; $j < $n ; $j ++) $sum += $mat [ $i ][ $j ] * $mat [ $i ][ $j ]; return floor (sqrt( $sum )); } // Returns trace of a // matrix of size n x n function findTrace( $mat , $n ) { $sum = 0; for ( $i = 0; $i < $n ; $i ++) $sum += $mat [ $i ][ $i ]; return $sum ; } // Driver Code $mat = array ( array (1, 1, 1, 1, 1), array (2, 2, 2, 2, 2), array (3, 3, 3, 3, 3), array (4, 4, 4, 4, 4), array (5, 5, 5, 5, 5)); echo "Trace of Matrix = " , findTrace( $mat , 5), "\n" ; echo "Normal of Matrix = " , findNormal( $mat , 5) ; // This code is contributed by anuj_67. ?> |
Output :
Trace of matrix = 15 Normal of matrix = 16
Time Complexity : O(n*n)
Space Complexity : O(1)
References:
https://en.wikipedia.org/wiki/Trace_(linear_algebra)
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