Matrices >

An idempotent matrix is one which, when multiplied by itself, doesn’t change. If a matrix A is idempotent, A^{2} = A.

## Examples of Idempotent Matrix

The simplest examples of n x n idempotent matrixes are the identity matrix I_{n}, and the null matrix (where every entry on the matrix is 0).

Nontrivial examples of 2 x 2 matrices are easy to come up with. Since A^{2} = A, we know that for a matrix ,

a = a^{2} + bc

b = ab + bd, so b – ab – bd = 0 and b(1 – a – d) = 0 and either b = 0 or d = 1 – a

c = ca + cd, so c – ca – cd = 0 and c(1 – a – d) = 0 and either c = 0 or d = 1 – a

d = bc + d^{2}.

To come up with your own idempotent matrix, start by choosing any value of a. For an example, let’s choose 3. Make d = 1 – a, or -2. Now -2 = bc + 4, by our a last equation above, so -6 = bc. As a quick check, see that this fits our first equation, a = 3 = a^{2} + bc = 9 – 6.

We can choose any values of b and c that fit bc = -6, so to make it easy on ourselves and stick to whole integers; b = -2 and c = 3. So the matrix

is idempotent.

You can come up with a similar set of algebra rules for finding idempotent matrices in any size n x n ; the larger the matrix gets, the stickier the algebra may become, but the method remains the same.

## Properties of Idempotent Matrices

Except for the identity matrix (I), every idempotent matrix is singular. What this means is that it is a square matrix, whose determinant is 0.

Since

[I – M] [I – M] = I – M – M + M^{2} = I – M – M + M = I – M,

the identity matrix minus any other idempotent matrix is also an idempotent matrix.

## The Idempotent Matrix in Statistics

Idempotent matrices are important in regression analysis and the theory of linear statistical models, especially as they relate to the analysis of variance and the theory of least squares.

## References

Harville D.A. (1997) Idempotent Matrices. In: Matrix Algebra From a Statistician’s Perspective. Retrieved December 4, 2017 from: from

https://link.springer.com/chapter/10.1007/0-387-22677-X_10

**Need help with a homework or test question?** With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you rather get 1:1 study help, Chegg Tutors offers 30 minutes of free tutoring to new users, so you can try them out before committing to a subscription.

If you prefer an **online interactive environment** to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

**Comments? Need to post a correction?** Please post a comment on our *Facebook page*.

Check out our updated Privacy policy and Cookie Policy