grandes-ecoles 2023 Q1

grandes-ecoles · France · mines-ponts-maths1__pc Groups Symplectic and Orthogonal Group Properties
Show that a matrix $S \in S_n(\mathrm{R})$ belongs to $S_n^+(\mathrm{R})$ if, and only if, $\mathrm{Sp}(S) \subset \mathbf{R}_+$.
Similarly, we will admit in the rest of the problem that: $S \in S_n^{++}(\mathrm{R})$ if, and only if, $\operatorname{Sp}(S) \subset \mathbf{R}_+^\star$. 
Show that a matrix $S \in S_n(\mathrm{R})$ belongs to $S_n^+(\mathrm{R})$ if, and only if, $\mathrm{Sp}(S) \subset \mathbf{R}_+$.

Similarly, we will admit in the rest of the problem that: $S \in S_n^{++}(\mathrm{R})$ if, and only if, $\operatorname{Sp}(S) \subset \mathbf{R}_+^\star$.
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