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 $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ and $Q^{-1} \cdot M \cdot Q = \operatorname{Diag}(M_1, M_2)$. Show that there exists $\rho _ { 3 } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { 3 } \leqslant \rho _ { 2 }$ and matrices $R _ { 1 } \in \mathrm { GL } _ { d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right) , R _ { 2 } \in \mathrm { GL } _ { n - d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right)$ such that the matrix $Q \cdot \operatorname { Diag } \left( R _ { 1 } , R _ { 2 } \right)$ is orthogonal. (One may use the result of question 17.)
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 $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ and $Q^{-1} \cdot M \cdot Q = \operatorname{Diag}(M_1, M_2)$.
Show that there exists $\rho _ { 3 } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { 3 } \leqslant \rho _ { 2 }$ and matrices $R _ { 1 } \in \mathrm { GL } _ { d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right) , R _ { 2 } \in \mathrm { GL } _ { n - d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right)$ such that the matrix $Q \cdot \operatorname { Diag } \left( R _ { 1 } , R _ { 2 } \right)$ is orthogonal. (One may use the result of question 17.)