the following problem has been keeping me up at night for the past two days. Any help or hint would be largely appreciated.
Let $B$, $C$ be matrices over the complex field ( $B$ is $k\times k$ and $C$ is $l\times l$). Let $D$ be any $k\times l$ matrix and $O$ be the zero $l\times k$ matrix. Furthermore I am given that $B$ and $C$ have no eigenvalues in common.
I need to show similarity between the following two matrices
\begin{pmatrix} B & D \\ O & C \end{pmatrix} and \begin{pmatrix} B & O \\ O & C \end{pmatrix}
So far I have got that both of them have the same characteristic polynomial that factors as the product of the characteristic polynomials of $B$ and $C$. Furthermore since $B$ and $C$ share no eigenvalues their respective Jordan canonical forms can be split into two families of blocks (with the eigenvalues of $B$ and $C$ respectively).
What I would like to argue (if possible) is that these blocks are equal and therefore the above two matrices have the same Jordan form.
Thaks in advance for any help provided
Typically this is proved as a corollary to a result by WE Roth:
The equations AX − YB = C and AX − XB = C in matrices Proc. Amer. Math. Soc., 3 (1952), pp. 392-396
Specifically, theorem 2 in the paper. Then you just need to show that if the characteristic polynomial of $B$ is relatively prime to that of $C$ the (linear) mapping sending $X$ to $BX - XC$ is an isomorphism (Using the matrices in your question).
For a detailed proof, see Proposition 5.20 in this dissertation: Sums and products of square-zero matrices.