Show that a matrix $S \in S _ { n } ( \mathbf { R } )$ belongs to $S _ { n } ^ { + } ( \mathbf { R } )$ if, and only if, $\operatorname { Sp } ( S ) \subset \mathbf { R } _ { + }$.
Show that a matrix $S \in S _ { n } ( \mathbf { R } )$ belongs to $S _ { n } ^ { + } ( \mathbf { R } )$ if, and only if, $\operatorname { Sp } ( S ) \subset \mathbf { R } _ { + }$.