TITLE: Showing two matrix blocks are similar
QUESTION [7 upvotes]: 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 $S^{-1} H S=K$.

REPLY [9 votes]: Using Jordan form: there exists matrices $P,Q,R$ such that $P^{-1}AP = J_A$, $Q^{-1}BQ = J_B$, and $RCR^{-1}  = J_C$ are all in Jordan normal form.
We then note that
$$
\pmatrix{P\\&Q}^{-1}\pmatrix{A\\&B}  \pmatrix{P\\&Q} = \pmatrix{J_A\\& J_B}\\
\pmatrix{P\\&R}^{-1}\pmatrix{A\\&C}  \pmatrix{P\\&R} = \pmatrix{J_A\\& J_C}
$$
By the uniqueness of Jordan form (up to permutations of blocks), the two matrices on the right can only be similar if $J_B$ is similar to $J_C$, which is to say that $B$ is similar to $C$ as desired.