We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$, and every $M \in \mathcal{C}_J$ has the form $M = \left( \begin{array}{cc} U & -V \\ V & U \end{array} \right)$ for some $U, V \in \mathcal{M}_m(\mathbb{R})$. Deduce that, for every matrix $M \in \mathcal { C } _ { J } , \operatorname { det } ( M ) \geqslant 0$. One may consider the product of block matrices $\left( \begin{array} { c c } I _ { m } & 0 \\ \mathrm { i } I _ { m } & I _ { m } \end{array} \right) \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) \left( \begin{array} { c c } I _ { m } & 0 \\ - \mathrm { i } I _ { m } & I _ { m } \end{array} \right)$.
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$, and every $M \in \mathcal{C}_J$ has the form $M = \left( \begin{array}{cc} U & -V \\ V & U \end{array} \right)$ for some $U, V \in \mathcal{M}_m(\mathbb{R})$. Deduce that, for every matrix $M \in \mathcal { C } _ { J } , \operatorname { det } ( M ) \geqslant 0$.
One may consider the product of block matrices $\left( \begin{array} { c c } I _ { m } & 0 \\ \mathrm { i } I _ { m } & I _ { m } \end{array} \right) \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) \left( \begin{array} { c c } I _ { m } & 0 \\ - \mathrm { i } I _ { m } & I _ { m } \end{array} \right)$.