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 = 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 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 A2= 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 An = 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 = 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.
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