Idempotent Matrix

An idempotent matrix is defined as a square matrix that, when multiplied by itself, results in the original matrix.

Consider a square matrix "P" of any order, and the matrix P is said to be an idempotent matrix if and only if P² = P.

• Idempotent matrices are singular and can have non-zero entries.
• Every identity matrix is also an idempotent matrix, as the identity matrix gives the same matrix when multiplied by itself.

Examples of Idempotent Matrix

The matrix given below is an idempotent matrix of order "2 × 2."

A_{22} = \left[\begin{array}{cc} 4 & -1\\ 12 & -3 \end{array}\right]

The matrix given below is an idempotent matrix of order "3 × 3."

B_{33} = \left[\begin{array}{ccc} 2 & -3 & -5\\ -1 & 4 & 5\\ 1 & -3 & -3 \end{array}\right]

Idempotent Matrix Formula

Let us consider a "2 × 2" square matrix: P = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right].

As P is an idempotent matrix, P² = P.
\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]\times\left[\begin{array}{cc} a & b\\ c & d \end{array}\right] = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]

\left[\begin{array}{cc} a^{2}+bc & ab+bd\\ ac+cd & bc+d^{2} \end{array}\right] = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]

Now, comparing the terms on each side, we get

• a² + bc = a
  bc = a − a²

• ab + bd = b
  ab + bd − b = 0
  b (a + d − 1) = 0
  b = 0 or a + d − 1 = 0
  d = 1 − a

So, if a matrix P = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] is said to be an idempotent matrix, if bc = a − a² and d = 1 − a.

Properties of Idempotent Matrix

The following are some important properties of an idempotent matrix:

• Every idempotent matrix is a square matrix.
• All idempotent matrices are singular matrices, apart from the identity matrix.
• The determinant of an idempotent matrix is either one or zero.
• The non-diagonal entries of an idempotent matrix can be non-zero entries.
• The trace of an idempotent matrix is always an integer and equal to the rank of the matrix.
• The eigenvalues of an idempotent matrix are either zero or one.
• The following is the relationship between idempotent and involutory matrices:

A square matrix "A" is said to be an idempotent matrix if and only if P = 2A − I is an involutory matrix. 

Where the Involutory matrix is, which P = P^-1.

Related Articles

• Minors and Cofactors of Determinants
• Determinant of a Matrix
• Adjoint of a Matrix

Solved Examples on Idempotent Matrix

Example 1: Verify whether the matrix given below is idempotent or not.

P = \left[\begin{array}{cc} 3 & 2\\ -3 & -2 \end{array}\right]

Solution:

To prove that the given matrix is idempotent, we have to prove that P² = P.

P^{2} = \left[\begin{array}{cc} 3 & 2\\ -3 & -2 \end{array}\right] \times\left[\begin{array}{cc} 3 & 2\\ -3 & -2 \end{array}\right]

P^{2} = \left[\begin{array}{cc} 9-6 & 6-4\\ -9+6 & -6+4 \end{array}\right]

P^{2} = \left[\begin{array}{cc} 3 & 2\\ -3 & -2 \end{array}\right] = P

Hence, verified.

So, the given matrix P is an idempotent matrix.

Example 2: Verify whether the matrix given below is idempotent or not.

B = \left[\begin{array}{ccc} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{array}\right]

Solution:

To prove that the given matrix is idempotent, we have to prove that B² = B.

B^{2} = \left[\begin{array}{ccc} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{array}\right] \times\left[\begin{array}{ccc} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{array}\right] 

B^{2} = \left[\begin{array}{ccc} (4+2-4) & (-4-6+8) & (-8-8+12)\\ (-2-3+4) & (2+9-8) & (4+12-12)\\ (2+2-3) & (-2-6+6) & (-4-8+9) \end{array}\right] 

B^{2} = \left[\begin{array}{ccc} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{array}\right]= B

Hence, verified.

So, the given matrix B is an idempotent matrix.

Example 3: Give an example of an idempotent matrix of order 2 × 2.

Solution:

We know that a matrix A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]   is said to be an idempotent matrix, if bc = a − a² and d = 1 − a.

Let us consider that a = 5

We have, d = 1 − a

d = 1 − 5 = −4

bc = a − a²

bc = 5 − 25 = −20

Now, let b = 4 and c = −5

So, the matrix is A = \left[\begin{array}{cc} 5 & 4\\ -5 & -4 \end{array}\right]

Example 4: Prove that an identity matrix is an idempotent matrix.

Solution:

To prove that the given matrix is idempotent, we have to prove that I² = I.

Let us consider an identity matrix of order 2 × 2, i.e., I = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]

I^{2} = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]\times\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]

I^{2} = \left[\begin{array}{cc} (1\times1+0\times0) & (1\times0+0\times1)\\ (0\times1+1\times0) & (0\times0+1\times1) \end{array}\right]

I^{2} = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right] = I

Hence, proved.

So, an identity matrix is an idempotent matrix.