Two linear maps: invertible case We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$. In this question, we assume that $M$ is invertible. a) Prove that $m = n$ and that $A$ and $B$ are invertible. b) Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form $$\begin{pmatrix} 0_r & B_1 \\ A_1 & 0_r \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{pmatrix},$$ where $A_1 = \mathrm{I}_r$ and $B_1 = \lambda \mathrm{I}_r + J_r$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.
\textbf{Two linear maps: invertible case}\\
We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$. In this question, we assume that $M$ is invertible.
a) Prove that $m = n$ and that $A$ and $B$ are invertible.
b) Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form
$$\begin{pmatrix} 0_r & B_1 \\ A_1 & 0_r \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{pmatrix},$$
where $A_1 = \mathrm{I}_r$ and $B_1 = \lambda \mathrm{I}_r + J_r$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.