A square matrix A is said to be idempotent if A^2 = A.

Question

Let A be an idempotent matrix.

(a) Show that I − A is also idempotent.

(b) Show that if A is invertible, then A = I.

(c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)

Answer 1

a)

Let B = (I-A)

B²=BB=(I-A)(I-A) = I - 2A + A² = I-2A+A = I-A = B

b)

A²=A

AA=A

A=A^-1A or A=AA^-1

A=I

c)

Ax=λx

A²=A

A²-A=0

(A²-A)x=0

A²x-Ax=0

λ²x-λx=0

(λ²-λ)=0

λ(λ-1)=0

λ= 0,1