We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$. By Theorem 1, there exists $\rho _ { 1 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 1 } \leqslant \rho$ such that $\chi$ factors in the form $\chi = F G$ with $F \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Only in this question, we assume that $d = n$. Show that there exists a symmetric matrix $M _ { 0 } \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$ such that $M = \lambda I _ { n } + t M _ { 0 }$ for all $t \in U _ { \rho _ { 1 } }$.