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 $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$. Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.
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 $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.