Let $S \in \mathcal{S}_{n}(\mathbb{R})$. a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$. b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\mathrm{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.
Let $S \in \mathcal{S}_{n}(\mathbb{R})$.
a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$.
b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\mathrm{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.