**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 *

*A*^{2}*=*In other words, an

**A.**Where**A**is n x n square matrix.**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.,

**A**=

^{2}**A )**then that matrix will be an Idempotent matrix.

Here if we observe the definition **A**** ^{2}**=

**A,**i.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**.

- If any Idempotent matrix is an identity matrix
**[I]**, then it will be a non-singular matrix. - When any Idempotent matrix
**[A]**is subtracted from identity matrix**[I],**then the resultant matrix**[I-A]**will also be an**idempotent matrix**. - 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.
- If a matrix
**[A]**is an idempotent matrix, then for all positive integer values of variable ‘**n**‘, the result**A**=^{n}**A**will always be true. - The Eigen-values of any Idempotent matrix will always be either
**0**or**1.**That means an idempotent matrix is always diagonalizable. - The trace of an idempotent matrix will be equal to the rank of that Idempotent matrix, hence trace will always be an integer value.
**For any 2 x 2 idempotent matrix [A].**

**a = a**^{2}+ 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 = d**^{2}+ 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., ** A^{2 }**], and then check whether the square of matrix [

**] gives the resultant matrix as the same matrix**

*A*^{2}**[A]**or not, (i.e,

*A*^{2}*=*If this condition is satisfied then given matrix will be an

**A).****idempotent matrix**otherwise it will not be an idempotent matrix.

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