We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M \in S _ { n } ( \mathbb { R } )$. We study the set $$\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\} .$$ We first study the case $n = 2$. We then denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \right)$. Show that $\mathcal { D } _ { M }$ is the line segment in $\mathbb { R } ^ { 2 }$ whose endpoints are $( \lambda _ { 1 } , \lambda _ { 2 } )$ and $( \lambda _ { 2 } , \lambda _ { 1 } )$.
We consider the application
$$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$
Let $M \in S _ { n } ( \mathbb { R } )$. We study the set
$$\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\} .$$
We first study the case $n = 2$. We then denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \right)$.\\
Show that $\mathcal { D } _ { M }$ is the line segment in $\mathbb { R } ^ { 2 }$ whose endpoints are $( \lambda _ { 1 } , \lambda _ { 2 } )$ and $( \lambda _ { 2 } , \lambda _ { 1 } )$.