Simple matrix equation I believe I'm missing an important concept and I need your help.
I have the following question:
"If $A^2 - A = 0$ then $A = 0$ or $A = I$"
I know that the answer is FALSE (only because someone told me) but when I try to find out a concrete matrix which satisfies this equation (which isn't $0$ or $I$) I fail.
Can you please give me a direction to find a concrete matrix? What is the idea behind this question?
Guy

Any projection operator obeys this relation.  It should be intuitive that when you apply this operator again, the projected vector should not change.
One can prove this concretely.  For a unit vector $u$, let $\underline A(a) = a - (u \cdot a)u$. This projects the vector $a$ onto the subspace orthogonal to $u$.  Clearly $\underline A^2(a) = a - (u \cdot a) u - (u \cdot a) u + (u \cdot u)(u \cdot a) u = a - (u \cdot a)u = \underline A(a)$.
