For $T = \left( t _ { i j } \right) \in \mathcal { M } _ { n + 1 } ( \mathbb { R } )$, we denote by $T _ { \leqslant n }$ the extracted matrix of size $n$ whose coefficients are the $t _ { i j }$ for $1 \leqslant i , j \leqslant n$. Let $M \in S _ { n + 1 } ( \mathbb { R } )$, and denote $\mathcal { C } _ { M } = \left\{ \operatorname { Sp } \left( \left( U M U ^ { - 1 } \right) _ { \leqslant n } \right) \in \mathbb { R } ^ { n } , \text { for } U \text { ranging over } O _ { n + 1 } ( \mathbb { R } ) \right\}$. We further assume that the eigenvalues of $M$ are distinct. We thus have $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$.
(a) Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right)$ such that $\operatorname { Sp } ( M )$ and $\widehat { \mu }$ are strictly interlaced. Show that $\widehat { \mu }$ belongs to $\mathcal { C } _ { M }$.
(b) Show that $$\mathcal { C } _ { M } = \left\{ \widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) , \text { such that } \operatorname { Sp } ( M ) \text { and } \widehat { \mu } \text { are interlaced } \right\} .$$