Let $A \in M_n$ and $B,C \in M_m$. Prove that if

$$H= \begin{bmatrix} A&0 \\ 0 & B \end{bmatrix}$$

is similar to

$$K = \begin{bmatrix} A&0 \\ 0 & C \end{bmatrix}$$

then $B$ is similar to $C$.

I am not sure how I would do this I know that if $H$ is similar to $K$ then for some non-singular matrix $S$ then $S^{-1} H S=K$.