Let $A , B , C , D$ be in $\mathcal { M } _ { m } ( \mathbb { R } )$ and let $M = \left( \begin{array} { l l } A & B \\ C & D \end{array} \right) \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ (block decomposition), where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $M \in \mathrm { Sp } _ { 2 m } ( \mathbb { R } )$ if and only if $$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$
Let $A , B , C , D$ be in $\mathcal { M } _ { m } ( \mathbb { R } )$ and let $M = \left( \begin{array} { l l } A & B \\ C & D \end{array} \right) \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ (block decomposition), where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $M \in \mathrm { Sp } _ { 2 m } ( \mathbb { R } )$ if and only if
$$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$