Two linear maps: decomposition 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 $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$. a) Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$. b) Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity. c) Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.
\textbf{Two linear maps: decomposition}\\
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 $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$.
a) Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$.
b) Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.
c) Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that
$$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$
and verify that these subspaces are stable under $H$.