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 $B = G ( M )$; we thus have $A , B \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 1 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$.

Show that there exist two matrices $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ such that:
\begin{itemize}
  \item $\operatorname { im } \left( B _ { 0 } U \right) = \operatorname { im } \left( B _ { 0 } \right)$,
  \item $\operatorname { im } \left( A _ { 0 } V \right) = \operatorname { im } \left( A _ { 0 } \right)$ and
  \item the block matrix $\left( B _ { 0 } U \mid A _ { 0 } V \right)$ is invertible.
\end{itemize}