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Program to find Normal and Trace of a matrix

  • Difficulty Level : Basic
  • Last Updated : 13 Apr, 2021

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)

 

 

Untitled

 



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.
?>

Javascript




<script>
 
// Javascript program to find trace
// and normal of given matrix
 
// Size of given matrix
var MAX = 100;
 
// Returns Normal of a matrix of size n x n
function findNormal(mat, n)
{
    var sum = 0;
    for(var i = 0; i < n; i++)
        for(var j = 0; j < n; j++)
            sum += mat[i][j] * mat[i][j];
             
    return parseInt(Math.sqrt(sum));
}
 
// Returns trace of a matrix of size n x n
function findTrace(mat, n)
{
    var sum = 0;
    for(var i = 0; i < n; i++)
        sum += mat[i][i];
         
    return sum;
}
 
// Driver code
var 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 ] ];
 
document.write("Trace of Matrix = " +
               findTrace(mat, 5) + "<br>");
document.write("Normal of Matrix = " +
               findNormal(mat, 5));
 
// This code is contributed by Kirti
 
</script>

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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