We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$, $$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c }
U & - V \\
V & U
\end{array} \right) .$$
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$,
$$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c }
U & - V \\
V & U
\end{array} \right) .$$