# Thread: Eigenvalue Proof

1. ## Eigenvalue Proof

Suppose that A is an n x n matrix with the property that A^2=A.
a) I have to show that if lambda is an eigenvalue of A, then lambda= 1 or 0.
b) Also I have to prove that A is diagonalizable.

I'm not sure how the property helps me find the eigenvalues of the matrix A.

2. If $Ax = \lambda x$ then $A^2x = A(Ax) = A(\lambda x) = \lambda^2x$. So if $A^2 = A$ and $x\ne0$ it follows that $\lambda^2 = \lambda$.