Here are two scenarios I would like to consider.
(1) $A^2 = A$
I think that if A has the Jordan form $A = PJP^{-1}$ then we know that $A^2 = A$ implies $J^2 = J$. So we know that the diagonal elements of J are either zero or one. Because $x^2 = x$ means $x = 0$ or $1$.
But what do we know about the off diagonal elements? The claim seems to be that all the diagonal elements must be zero otherwise we won't have $J^2 = J$. Could some one explain this in detail? It is not very clear to me.
(2) $A^2 = I$.
Under this condition, we know that $J^2 = I$. Again, following the similar logic, we can have that on the diagonal of J, it must be either 1 or -1. The question is what we know about the off-diagonal element? Why do all the off diagonal elements have to be zeros?
Thank you