Can we say that the matrix $\begin{bmatrix} A & A \\ 0 & A \end{bmatrix}$ is diagonalizable if and only if $A = 0$? Consider the block matrix 

$$\begin{bmatrix} A &amp; A \\ 0 &amp; A \end{bmatrix}$$

where $A$ is a $n\times n$ complex matrix. 

Can we say that this matrix is diagonalizable if and only if $A = 0$?
Let $m$ be the minimal polynomial of $A$ and $M=\begin{pmatrix}A&amp;A\\0&amp;A\end{pmatrix}$. Note that $m(M)=\begin{pmatrix}m(A)&amp;Am'(A)\\0&amp;m(A)\end{pmatrix}$. 

Thus $m^2(M)=0$ and, since $M$ is diagonalizable, $m(M)=0$. Suppose that $m$ has multiple roots; then, there is a polynomial $p$ that strictly divides $m$ and satisfies $p(M)=0$; that implies $p(A)=0$, a contradiction. 

Finally, $m$ has simple roots and $m(M)=0$ implies that $Am'(A)=0$; since $m'(A)$ is invertible, $A=0$.  $\square$
HINT: A matrix $B$ is diagonalizable if and only if $B$ is normal ($BB^*=B^*B$).