JavaScript Program to check idempotent matrix

An idempotent matrix is a square matrix that has the same number of rows and columns and when we multiply the matrix by itself the result will be equal to the same matrix. We will be given a matrix and we have to find that whether it is an idempotent matrix or not.

Mathematical Definition

If the given matrix is M, then for M to be an idempotent matrix it should follow the property:

M * M = M

Matrix Multiplication Overview

Multiplication of a matrix with another matrix produces another matrix and if the given matrix is the square matrix of N×N then the resultant matrix will also be of the same dimensions (N×N).

Each index (i,j) of the result matrix of the multiplication of two matrices A and B is the sum of the multiplicative addition of the jth column of matrix A with the ith row of matrix B.

Example: Identity Matrix

Input:

Mat = [ [1, 0],
        [0, 1]]

Output:

Yes, the given matrix is an Idempotent matrix.

Explanation:

For the cell (0,0): We have to multiply [1,0] with [1,0] => 1*1 + 0*0 = 1. 
For the cell (0,1): We have to multiply [1,0] with [0,1] => 1*0 + 0*1 = 0. 
For the cell (1,0): We have to multiply [0,1] with [1,0] => 0*1 + 1*0 = 0.
For the cell (1,1): We have to multiply [0,1] with [0,1] => 0*0 + 1*1 = 1.  
So, the final matrix we will get is: [ [1, 0], [0, 1]]

Algorithm Approach

We will implement the following steps to check if a matrix is idempotent:

• Create a function that takes the matrix as a parameter to check if it is idempotent

• Get the length of the matrix and traverse each cell using nested for loops

• At each index (i,j), calculate the value that would be present in the result matrix after multiplication

• Use a loop to compute the dot product of row i and column j

• Compare the computed value with the original matrix value at that position

• Return false if any values don't match, otherwise return true

Implementation

// function to check if the current matrix is an Idempotent matrix or not
function check(mat){

   // getting the size of the given matrix. 
   var n = mat.length;    
   
   // traversing over the given matrix 
   for(var i = 0; i < n; i++){
      for(var j = 0; j < n; j++){
      
         // for the current cell calculating the value present in the resultant matrix 
         
         // variable to store the current cell value 
         var cur = 0;
         for(var k = 0; k < n; k++){
            cur += mat[i][k] * mat[k][j];
         }
         
         // if the current value is not equal to value we got for this cell then return false 
         if(cur != mat[i][j]){
            return false;
         }
      }
   }
   
   // returning true as all the values matched 
   return true;
}

// defining the matrix 
var mat = [[2, -2, -4],
           [-1, 3, 4 ],
           [1, -2, -3]];

console.log("The given matrix is: ");
console.log(mat);

// calling the function to check if the current matrix is idempotent matrix or not 
if(check(mat)){
   console.log("The given matrix is idempotent matrix");
}
else {
   console.log("The given matrix is not idempotent matrix");
}

The given matrix is: 
[ [ 2, -2, -4 ], [ -1, 3, 4 ], [ 1, -2, -3 ] ]
The given matrix is idempotent matrix

Testing with Identity Matrix

// Testing with identity matrix (always idempotent)
var identityMatrix = [[1, 0, 0],
                      [0, 1, 0],
                      [0, 0, 1]];

console.log("Testing identity matrix:");
console.log(identityMatrix);

if(check(identityMatrix)){
   console.log("The identity matrix is idempotent matrix");
}
else {
   console.log("The identity matrix is not idempotent matrix");
}

Testing identity matrix:
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
The identity matrix is idempotent matrix

Time and Space Complexity

The time complexity of the above code is O(N³) where N is the number of rows of the given matrix. For each cell, we have to multiply the current row with the current column producing a factor of N, and there are a total of N² cells.

The space complexity of the above code is O(1), as we are not using any extra space to store the matrix.

Conclusion

In this tutorial, we implemented a JavaScript program to check if a given matrix is idempotent. An idempotent matrix satisfies the property M × M = M, and our solution efficiently verifies this by computing matrix multiplication and comparing results with the original matrix.