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 A2 = 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.
Also if square of any matrix gives same matrix ( i.e, A2 = A ) then that matrix will be Idempotent matrix.
Here if we observe the definition A2= 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 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 An = 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 = 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 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, A2 ] and then check that whether the square of matrix [A2] gives resultant matrix as same matrix [A] or not, (i.e, A2 = A). If this condition satisfies then given matrix will be idempotent matrix otherwise it will not be an idempotent matrix.