We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$, and $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ where $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ are as in question 20. Show that there exists $\rho _ { 2 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 2 } \leqslant \rho _ { 1 }$ such that $Q \in \operatorname { GL } _ { n } \left( \mathscr { D } _ { \rho _ { 2 } } ( \mathbb { R } ) \right)$. (One may use the result of question 6.)
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$, and $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ where $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ are as in question 20.
Show that there exists $\rho _ { 2 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 2 } \leqslant \rho _ { 1 }$ such that $Q \in \operatorname { GL } _ { n } \left( \mathscr { D } _ { \rho _ { 2 } } ( \mathbb { R } ) \right)$. (One may use the result of question 6.)