Idempotent Matrix

An idempotent matrix is one that when multiplied by itself produces the same matrix.

A = A

A is a n × n square matrix.

As a result, an idempotent matrix is one that does not change when multiplied by itself. If matrix A is idempotent, then

A2 = A

A3 = A

An = A

In reality, this sort of matrix gets its name because idempotence is a feature of some operations in mathematics that indicates the same result is always obtained regardless of how many times it is done.

Examples of idempotent matrix

Idempotent matrix examples can be of various forms. Some of them are as follows:

·       Idempotent matrix example for a 2 x 2 matrix

A= 

B= 

·       Idempotent matrix example for a 3 x 3 matrix

A=

B= 

Application of Idempotent Matrix

One of the most important uses of the idempotent matrix is its ease of use and utility in solving the [M] matrix and the Hat matrix during regression analysis in econometrics.

The idempotency of the [M] matrix is crucial in various regression analysis and econometric computations.

How do we know if a matrix is idempotent?

It is fairly simple to determine whether or not a given matrix [A] is an idempotent matrix.

Simply multiply the provided matrix [A] by the same matrix [A] to discover the square of the supplied matrix, i.e., [A2], and then check whether the square of the matrix yields the same matrix as [A] or not.

In other words, A2= A.

If this condition is met, the supplied matrix is an idempotent matrix; otherwise, it is not an idempotent matrix.

Conditions of Idempotent Matrix

To be an idempotent matrix, each 2 x 2 square matrix must be either a diagonal matrix of order 2 x 2 or have a trace value of 1.

The formula for a 2 x 2 idempotent matrix

The formula to obtain an idempotent matrix is given below. 

with a2 + bc = a

So the components of an idempotent matrix secondary diagonal can be any as long as the criterion a2+ bc = a is satisfied and the numbers of the main diagonal must be ay1-a.

In addition to all of the matrices specified by this formula, we must include the identity matrix of order 2, which, while not fulfilling the formula, is also an idempotent matrix. 

Properties of Idempotent Matrix

The important Idempotent matrix properties are as follows:

•  If an idempotent matrix [I] is discovered, it must be non-singular.

• If we subtract an idempotent matrix from an identity matrix, we get an idempotent solution matrix.

• If a non-identity matrix is also an idempotent matrix, the number of independent rows and columns is less than the total number of rows and columns.

• An idempotent matrix’s eigenvalues are always 0 and 1.

• A diagonalization of an idempotent matrix is impossible.

• An idempotent matrix’s rank is always equal to its trace. As a result, the trace is always an integer.

• For a idempotent matrix [A], if 

A=

 Then,

a= a2 + bc

b= ab + bc, which means that b(1-a-d)=0, i.e. either b= 0 or d= (1-a)

c= ac + dc, which means that c(1-a-d)=0, i.e. either c= 0 or d= (1-a)

d= d2 + bc

Conclusion

We now understand what an idempotent matrix is, as well as the idempotent matrix properties and idempotent matrix examples. We can now claim that idempotent matrices are significant in regression analysis and the theory of linear statistical models, particularly in relation to the variable analysis and the theory of least squares. If and only if it is the identity matrix, an idempotent matrix is also invertible.