Idempotent Matrix

Idempotent matrix is a square matrix which when multiplied by itself, gives back the same matrix. A matrix M is said to be an idempotent matrix if M² = M. Further every identity matrix can be termed as an idempotent matrix.

The idempotent matrix is a singular matrix and can have non-zero elements. Let us learn more about the properties of an idempotent matrix with examples, FAQs.

Idempotent matrix is a square matrix, which multiplied by itself, gives back the initial square matrix. A matrix M, when multiplied with itself, gives back the same matrix M, M² = M.

Let us consider a matrix A = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\). Further since A is taken as an idempotent matrix, we have A² = A.

\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) × \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\)

\(\begin{pmatrix}a^2+bc&ab+bd\\ac+cd&bc+d^2\end{pmatrix}\) = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\)

Here let us compare the terms on either sides.

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

From the above derivation we can understand that a matrix A =\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is an idempotent matrix if d = 1 - a, and bc = a - a². Further using these two conditions for a 2 x 2 square matrix, we can create an idempotent matrix. Let us create an idempotent matrix by taking a = 5, and we have the other elements of the matrix as follows.

d = 1 - a = 1 - 5 = -4

bc = a - a² = 5 - 5² = 5 - 25 = -20

bc = -20

The possible combinations for the values of b and c are b = 10, and c = -2. Hence one of the idempotent matrices which can be formed is as follows.

P = \(\begin{pmatrix}5&10\\-2&-4\end{pmatrix}\)

Also, all the identity matrices on multiplication with itself give back the identity matrix, and hence the identity matrix is also considered an idempotent matrix.

I_{2 x 2}= \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)

I_3x3 = \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)

The determinant of an idempotent matrix is always equal to zero, and hence an idempotent matrix is also a singular matrix.

The following are some of the important properties of an idempotent matrix.

• The idempotent matrix is a square matrix.
• The idempotent matrix has an equal number of rows and columns.
• The idempotent matrix is a singular matrix
• The non-diagonal elements can be non-zero elements.
• The eigenvalues of an idempotent matrix is either 0 or 1.
• The trace of an idempotent matrix is equal to the rank of a matrix
• The trace of an idempotent matrix is always an integer.

Related Topics

The following topics help in a better understanding of the idempotent matrix.

• Nilpotent Matrix
• Transformation Matrix
• Matrix Operations
• Non-Singular Matrix
• Unitary Matrix

What Is Idempotent Matrix?

Idempotent matrix is a square matrix which when multiplied by itself, gives back the same matrix. A matrix M is said to be an idempotent matrix if M² = M. Further every identity matrix can be termed as an idempotent matrix.

What Is The Order Of An Idempotent Matrix?

The idempotent matrix has an order of the form n x n. The idempotent matrix is a square matrix with an equal number of rows and columns, and generally, the idempotent matrix is of the order 2 x 2, or 3 x 3.

What Are The Properties Of an Idempotent Matrix?

The three important properties of idempotent matrices are as follows.

• The idempotent matrix is a singular matrix.
• The eigenvalues of an idempotent matrix is either 0 or 1.
• The trace of an idempotent matrix is equal to the rank of a matrix.