# Idempotent Matrix – Definition, Example, Properties

Contents

Idempotent matrix is a matrix when multiplied by itself will produce itself (same matrix). It is an idempotent matrix if A x A = A, where A is an n x n matrix.

## The Definition of Idempotent Matrix

Just as stated above, matrix A is an idempotent matrix if we multiply it with itself and produces the very same matrix, or in n x n matrix:

Thus,

Why is it known as “idempotent”?

It is called idempotent because of its property:

Idempotence is a property when a computer or mathematical operation is executed or multiplied several times and it still produces the same result.

## Example

Idempotent matrix comes with several forms, it can be 2 x 2 matrix, 3 x 3 matrix, 4 x 4 matrix, and so on.

For a 2 x 2 matrix:

For a 3 x 3 matrix:

## How to Check if it is an Idempotent Matrix?

Well, from the example above, you should have understood how to prove whether a matrix is an idempotent matrix or not.

Say, you have Matrix A (doesn’t matter whether it is 2×2, 3×3, or more).

Next, just multiply Matrix A with Matrix A.

If the result is exactly the same as Matrix A, we can conclude that Matrix A is an idempotent matrix.

Hence,

Otherwise, if the result is not exactly the same as matrix A, we can conclude that Matrix A is not an idempotent matrix.

It is quite similar to a nilpotent matrix.

## Formula

The formula we need to verify an idempotent matrix can be seen above, now we will try to make an idempotent matrix.

Assume we want to make a 2×2 idempotent matrix

If we want to make it, we can put any number for a-b-c-d but we have to make sure that:

As long as this rule is fulfilled, other numbers don’t matter.

Or we can use

Using a simple 2×2 matrix, if the matrix above is idempotent, then

## Properties of Idempotent Matrix

What are the important things to remember about an idempotent matrix? There are several properties of an idempotent matrix, such as:

1. Identity matrix is an idempotent matrix.
2. An idempotent matrix is a nonsingular matrix.
3. For a non-identity idempotent matrix, its independent rows (and columns) number is less than its rows (and columns) number.
4. If an idempotent matrix is subtracted from an identity matrix, the result is also idempotent.
5. The eigenvalues of an idempotent matrix is always 0 or 1.
6. The determinant of an idempotent matrix is always 0 or 1.
7. An idempotent matrix is always diagonalizable.
8. The trace of an idempotent matrix (the sum of the values on its main diagonal) is equal to its rank of the matrix and always an integer.

How do you check if a matrix is Idempotent or not?

Idempotent matrix is a matrix when multiplied by itself will produce itself (same matrix). It is an idempotent matrix if A x A = A, where A is an n x n matrix.

What is an example of an idempotent matrix?

The easiest example of an idempotent matrix is an identity matrix where it only contains 1 on its main diagonal and 0 for the rest.

What is an idempotent matrix formula?

An idempotent matrix is a square matrix that we multiply with itself and produces the very same matrix. A matrix is said idempotent only if A x A = A.

Is an idempotent matrix always 0 or 1?

The determinant of an idempotent matrix is always either zero or one.