Diagonalizable transmit to submatrix

Question

If $$\begin{pmatrix} A & B\\ \Large 0 & C \end{pmatrix}$$ is similar to a diagonal matrix, are $A$ and $C$ also similar to diagonal matrices?

Answer 1

As the comment provided indicates, the answer is yes. The quickest way to see this is to note that a matrix is diagonalizable if and only if its minimal polynomial has no repeated roots.

Let $m$ be the minimal polynomial of the block matrix $M$. Of course, $m(M) = 0$. However, by block-matrix multiplication, we see that $$ m(M) = \pmatrix{ m(A) & \star\\ 0 & m(C)} $$ so, we deduce that $m(A) = 0$ and $m(C) = 0$. So, the minimal polynomials of $A$ and $C$ divide $m$. So, the minimal polynomials of $A$ and $C$ have no repeated roots. So, both $A$ and $C$ are diagonalizable.