If a block upper triangular matrix is diagonizable, then are some of its blocks diagonizable?

Question

Suppose $n>1$ and $X, Y, Z \in M_n\left(\mathbb{C}\right)$. Let $A = \left(\begin{matrix}X & Y\\0_n & Z\end{matrix}\right)$. If $A$ is diagonalizable, then are $X$ and $Z$ also diagonalizable?

Answer 1

If $A$ is diagonalizable then its minimal polynomial $m_A$ has roots of algebraic multiplicity $1$.

Now powers of $A$ are upper-triangular with powers of $Z$ and $Z$ on the diagonal: $$ \pmatrix{X&Y\\0&Z}\pmatrix{X&Y\\0&Z} = \pmatrix{X^2&*\\0&Z^2}. $$ Per induction you can show that for every polynomial $p$ it holds $$ p(A) = \pmatrix{p(X)&*\\0&p(Z)}. $$ It follows that $m_A(X)=0$ and $m_A(Z)=0$. This implies diagonalizability of $X$ and $Y$ by the properties of $A$.