An idempotent matrix is one which, when multiplied by itself, doesn’t change. If a matrix A is idempotent, A2 = A.
Examples of Idempotent Matrix
The simplest examples of n x n idempotent matrices are the identity matrix In, and the null matrix (where every entry on the matrix is 0).
Nontrivial examples of 2 x 2 matrices are relatively easy to come up with (Need help? Check out our tutoring page!). Since A2 = A, we know that for a matrix ,
a = a2 + 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 + d2.
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 = a2 + 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
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
[I – M] [I – M] = I – M – M + M2 = I – M – M + M = I – M,
the identity matrix minus any other idempotent matrix is also an idempotent matrix.
The Idempotent Matrix in Statistics
Harville D.A. (1997) Idempotent Matrices. In: Matrix Algebra From a Statistician’s Perspective.