# Find normal and Trace of a matrix in C++

Hello, in this tutorial, we will learn how to find normal and the trace of a matrix with a solved example, algorithm, and C++ program.

Normal of a matrix is the square root of the sum of the squares of each element in that matrix.

Trace of a square matrix is the sum of the primary diagonal elements.

## How to find normal and the trace of a matrix in C++

let’s take an example and solve it manually to understand it in a better way.

Example

```consider a 4*4 matrix
input matrix:
1 2 3 4
5 6 7 8
9 0 1 2
3 4 5 6

finding normal and trace of the matrix:
normal = sqrt(1*1 + 2*2 + 3*3 + 4*4 + 5*5 + 6*6 + 7*7 + 8*8 + 9*9 + 0*0 + 1*1 +
2*2 + 3*3 + 4*4 + 5*5 + 6*6)
= sqrt(1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 0 + 1 + 4 + 9 + 16 +
25 + 36)
= sqrt(376)
= 19.39

Trace = 1 + 6 + 1 + 6 = 14

Therefore, normal and Trace of input matrix are 376 and 14 respectively.```

Approach:

To find the normal of a matrix, traverse the matrix form starting to end. Add the square of each element of the matrix and display it. To find the Trace, traverse the matrix, add the diagonal element and display the addition.

### Algorithm to find normal and the trace of a matrix

1. Declare and initialize a matrix of size n*n.
2. declare two variable consider double and trace and initialize it with zero.
3. Traverse the matrix from matrix[i][j] to matrix[n-1][n-1], for each matrix[i][j] add the into double.
4. for each matrix[i][j], when i is equal to j, add each matrix[i][j] to the variable trace.
5. display square root of double and the value of trace.

## C++ program to find normal and the trace of a matrix

```#include<bits/stdc++.h>
using namespace std;

//Function to find Normal of a matrixrix of size n x n(m)
float Normal(int matrix, int n)
{
int sum = 0;

for (int i=0; i<n; i++)
for (int j=0; j<n; j++)
sum += matrix[i][j]*matrix[i][j];

return sqrt(sum);
}

//Function to find trace of a matrixrix of size n x n(m)
int Trace(int matrix, int n)
{
int sum = 0;

for (int i=0; i<n; i++)
sum += matrix[i][i];

return sum;
}

// Driven source
int main()
{
int matrixrix, n, m;

//taking input in m and n for length of row and column
cout<<"Enter the length of row of the matrix: ";
cin >> m;
cout<<"Enter the length of column of the matrix: ";
cin >> n;

if(m != n){
cout<<"\nlength of row and column must be same.";
exit(0);
}

//taking input in the matrixrix
cout << "\nInput the element of matrix: \n";
for(int i = 0; i < m; i++)
for(int j = 0; j < n; j++)
cin>>matrixrix[i][j];

cout << "\nTrace of Matrix = "
<< Trace(matrixrix, n) << endl;
cout << "Normal of Matrix = "
<< Normal(matrixrix, n) << endl;
return 0;
}```

Output

```Enter the length of row of the matrix: 4
Enter the length of column of the matrix: 4

Input the element of matrix:
1 2 3 4
5 6 7 8
9 0 1 2
3 4 5 6

Trace of Matrix = 14
Normal of Matrix = 19.3907```

#### Time complexity:

O(n^2) where n is the length of row and column of the matrix.