A matrix A is idempotent if A2 = A. Show that the only possible eigenvalues of an idempotent matrix are λ = 0 and λ = 1. Then give an example of a matrix that is idempotent and has both of these two values as eigenvalues

Suppose that λ is an eigenvalue of A. Then there is an eigenvector x, such that Ax = λx. We have,

λx = Ax                                                            x eigenvector of A

= A²x                                                           A is idempotent

= A(Ax)

= A(λx)                                                       x eigenvector of A

= λ(Ax)                                                      Theorem MMSMM

= λ(λx)                                                       x eigenvector of A

= λ²x

From this we get

0 = λ²x − λx

= (λ² − λ)x           Property DSAC

Since x is an eigenvector, it is nonzero, and Theorem SMEZV leaves us with the conclusion that λ²   λ = 0, and the solutions to this quadratic polynomial equation in λ are λ = 0 and λ = 1.

The matrix

is idempotent (check this!) and since it is a diagonal matrix, its eigenvalues are the diagonal entries, λ = 0 and λ = 1, so each of these possible values for an eigenvalue of an idempotent matrix actually occurs as an eigenvalue of some idempotent matrix. So we cannot state any stronger conclusion about the eigenvalues of an idempotent matrix, and we can say that this theorem is the “best possible.”