Let $A$ be a $3 \times 3$ complex matrix with two distinct eigenvalues $e_1, e_2$. Write all possible Jordan canonical forms of $A$ and find the spectrum of $A$, in case(1) $A^{2}=A$ and case(2) $A^{2}=I$
So I know that the characteristic polynomial in $\mathbb{C}$ is always fully reducible, which implies that there are three eigenvalues for A, right? Then $e_3$ is either equal to $e_1$ or $e_2$. Then to find each $J_{i}$, would I find the kernel of each eigenvalue?