# Idempotent Matrix – Definition, Examples and its Properties

What is an Idempotent Matrix?

Idempotent Matrix: Mathematically we can define an Idempotent matrix as A square matrix [A] will be called an Idempotent matrix if and only if it satisfies the condition  A2 = AWhere is n x n square matrix.  In other words, an Idempotent matrix is a square matrix which when multiplied by itself, gives result as same square matrix. Also if the square of any matrix gives the same matrix (i.e., A2 = A )  then that matrix will be an Idempotent matrix.

Here if we observe the definition   A2Ai.e., A = square of (A). It means we can say that the Idempotent matrix [A] is always the square of the same matrix [A].

Examples of Idempotent Matrix

Example of 2 x 2 Idempotent Matrix

Example of 3 x 3 Idempotent Matrix

Conditions of Idempotent Matrix

The necessary conditions for any 2 x 2 square matrix to be an Idempotent matrix is that either it should be a diagonal matrix of order 2 x 2, or its trace value should be equal to 1.

## Properties of Idempotent Matrix

These are some important properties of the Idempotent matrix.

1. If any Idempotent matrix is an identity matrix [I], then it will be a non-singular matrix.
2. When any Idempotent matrix [A] is subtracted from identity matrix [I], then the resultant matrix [I-A] will also be an idempotent matrix.
3. If a non-identity matrix is an idempotent matrix then its number of independent rows and columns will always be less than the number of total rows and columns of that Idempotent matrix.
4. If a matrix [A] is an idempotent matrix, then for all positive integer values of variable ‘n‘, the result An = will always be true.
5. The Eigen-values of any Idempotent matrix will always be either or 1. That means an idempotent matrix is always diagonalizable.
6. The trace of an idempotent matrix will be equal to the rank of that Idempotent matrix, hence trace will always be an integer value.
7. For any 2 x 2 idempotent matrix [A].
• a = a2 + bc
• b = ab + bc, implying that  b(1 – a – d) = 0, so either b = 0, or d = (1 – a)
• c = ac + dc, implying that  c(1 – a – d) = 0, so either c = 0, or d = (1 – a)
• d = d2 + bc

Application of Idempotent Matrix

One of the very important applications of the Idempotent matrix is that it is very easy and useful for solving the [ M ] matrix and Hat matrix during regression analysis and econometrics.

The idempotency of the [ M ]  matrix plays a very important role in other calculations of regression analysis and econometrics.

How do you know if a Matrix is Idempotent?

It is very easy to check whether a given matrix [A] is an idempotent matrix or not. Simply multiply that given matrix [A] with the same matrix [A] find the square of the given matrix [i.e., A2 ], and then check whether the square of matrix [A2] gives the resultant matrix as the same matrix [A] or not, (i.e, A2 = A). If this condition is satisfied then given matrix will be an idempotent matrix otherwise it will not be an idempotent matrix. 