Normal and Trace of a Matrix in Java

In this section, we will learn how to calculate the normal and trace of a matrix in Java. Before moving to, the program, first we will understand the what is normal and trace of a matrix.

Normal of a Matrix

The normal of a matrix is the square root of the sum of squares of all the elements of a matrix. For example, consider the following matrix.

First, we will calculate the sum of the square of each element.

9²+ 8² + 2² + 1² + 4² + 7² + 3² + 5² + 6²

81 + 64 + 4 + 1 + 16 + 49 + 9 + 25 + 36 = 285

Now, calculate the square root of the sum of squares.

√285 = 16.8819430161

Trace of a Matrix

The trace of a matrix is the sum of all the elements present in the principal diagonal (upper left to lower right). Note that the matrix must be a square matrix (the number of rows and columns must be the same). It is useful to prove results in linear algebra. For example, consider the following matrix.

Trace for the above matrix is 5 + 4 + 7 = 16.

Java Program to Find Normal and Trace of a Matrix

MatrixNormalTrace1.java

import java.util.*;
public class MatrixNormalTrace1
{
    public static void main(String args[])
    {
        int array[][] = new int[5][5];
        int i, j;
        double sum = 0, square = 0, result = 0;
  	Scanner s = new Scanner(System.in);
  	System.out.print("Enter the number of rows: ");
  	//reads number of rows from the user
  	int row = s.nextInt();
  	System.out.print("Enter the number of columns: ");
  	//reads number of columns from the user
  	int column = s.nextInt();
	System.out.println("Enter matrix:");
	//loop for rows
 	for(i = 0; i < row; i++)
  	{
  	    //loop for columns
   	    for(j = 0; j < column; j++) 
     	    {
     	        //reads the matrix elements 
      	        array[i][j] = s.nextInt();
      	        //prints space
                System.out.print(" ");
     	    }
        }
	System.out.println("You have entered the following matrix: ");
	//prints the matrix
 	for(i = 0; i < row; i++)
        {
      	    for(j = 0; j < column; j++)
            {
         	System.out.print(array[i][j]+" ");
            }
            System.out.println(" ");
    	}
        System.out.println("The trace of the given matrix is: ");
  	for(i = 0; i < row; i++)
  	{  
    	    for(j = 0; j < column; j++)
       	    {
       	        //checks if an element is of the same rows and column or not, if same calculates trace of the matrix
                if(i == j)
            	 {
            	     //find the sum of principal diagonal
               	     sum = sum + (array[i][j]);
               	 }
            }
        }
        //prints trace
        System.out.println(sum);  
        System.out.println("The normal of the given matrix is: "); 
        
   	for(i = 0; i < row; i++)
   	{
    	    for(j = 0; j < column; j++)
       	    {
       	        //calculates the sum of square of matrix elements 
       	        square = square + (array[i][j])*(array[i][j]);
            }
    	}
    	//calculates square root of the sum
        result = Math.sqrt(square);
        //prints normal of the given matrix
        System.out.println(result);
    }
}

Output:

Similarly, we can also find the normal and trace of a matrix using function.

MatrixNormalTrace2.java

import java.io.*;
public class MatrixNormalTrace2
{
//size of the given matrix
static int MAX = 10;
//the method calculates and returns Normal of a matrix 
static int calculateNormal(int matrix[][], int n)
{
	int sum = 0;
	for (int i=0; i<n; i++)
		for (int j=0; j<n; j++)
			sum = sum + matrix[i][j]*matrix[i][j];
	return (int)Math.sqrt(sum);
}
//the method calculates and returns trace of a matrix
static int calculateTrace(int matrix[][], int n)
{
	int sum = 0;
	for (int i=0; i<n; i++)
		sum = sum + matrix[i][i];
	return sum;
}
//driver code
public static void main (String args[]) 
    {
        //initializing matrix elements 
		int matrix[][] = {{6, 3, 2, 1, 4},
		                  {7, 8, 3, 12, 9},
		                  {1, 11, 10, 5, 8},
		                  {5, 18, 6, 9, 14},
		                  {19, 23, 3, 12, 11},
	                     };
	    System.out.println ("Trace of the give matrix is: "+ calculateTrace(matrix, 5));
	    System.out.println ("Normal of the given matrix is: "+ calculateNormal(matrix, 5));
	}
}

Output: