**What is an Idempotent matrix?**

**Definition:**** **Mathematically we can define

**as: a**

**Idempotent matrix***square matrix*

**[A]***will be called*

**if and only if it satisfies the condition****Idempotent matrix**

*A*^{2}*=*

**Where****A.****is n x n square matrix.****A**In other words, an ** Idempotent matrix** is a square matrix which when multiplied by itself, gives result as same square matrix.

Also if square of any matrix gives same matrix ** (** i.e,

**=**

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

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

**i.e,**

**A,****= square of**

**A****It means we can say that the Idempotent matrix**

**(A).****is always the square of same matrix**

**[A]****.**

**[A]****Examples of Idempotent matrix**

**Examples of Idempotent matrix**

**Example of 2 x 2 Idempotent matrix**

**Example of 2 x 2 Idempotent matrix**

**Example of 3 x 3 Idempotent matrix**

**Example of 3 x 3 Idempotent matrix**

**Conditions of 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**

**Properties of Idempotent matrix**

These are important properties of Idempotent matrix.

- If any Idempotent matrix is identity matrix
, then it will be non-singular matrix.**[I]** - When any Idempotent matrix
is subtracted from identity matrix**[A]**then the resultant matrix**[I],**will also be an idempotent matrix.**[I-A]** - 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
is an idempotent matrix, then for all positive integer values of variable ‘**[A]**‘, the result**n**=**A**^{n}will always true.**A** - The Eigen values of any Idempotent matrix will always be either
or**0**That means an idempotent matrix is always diagonalizable.**1.** - 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 idempotent matrix otherwise it will not be an idempotent matrix.

**A).****Also Read:**

**Notch filter- Theory, circuit design and Application**

**Notch filter- Theory, circuit design and Application**