Idempotent matrix is a square matrix which when multiplied by itself, gives result as same square matrix, and it satisfies the condition A2 = A.

## What is an Idempotent matrix?

**Definition:**** **Mathematically we can define **Idempotent
matrix** as: A* square matrix ***[A]** *will be called Idempotent matrix if and only if
it satisfies the condition *

*A*^{2}*=*

**A.**Where**A**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.

**[ ##eye## Involutory
matrix and its properties]**

Also if square of any matrix gives same matrix **(** i.e, **A ^{2}** =

**A )**then that matrix will be 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 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 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 Idempotent matrix**.

- If
any Idempotent matrix is identity matrix
**[I]**, then it will be 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 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 Idempotent matrix is
that it is very easy and useful for solving **[ M ] **matrix
and Hat matrix during **regression
analysis and econometrics**.

The idempotency of
**[ M ] **matrix plays 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 same matrix **[A]** and find the square of given matrix [ i.e,
** A^{2 }**] and then check that whether the square of
matrix [

**] gives resultant matrix as same matrix**

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

*A*^{2}*=*If this condition satisfies then given matrix will be

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

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Thanks for such an awesome article

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