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A square matrix A is said to be idempotent if A2 = A. Let A be...

A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.

  1. Show that I − A is also idempotent.
  2. Show that if A is invertible, then A = I.
  3. 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.)
  4. Let W = col(A). Show that TA(x) = projW x and TI−A(x) = projW ⊥ x.
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