Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $B \in S _ { n } ( \mathbf { R } )$. Show that there exists a diagonal matrix $D \in M _ { n } ( \mathbf { R } )$ and $Q \in G L _ { n } ( \mathbf { R } )$ such that $B = Q D Q ^ { \top }$ and $A = Q Q ^ { \top }$. What can be said about the diagonal elements of $D$ if $B \in S _ { n } ^ { + + } ( \mathbf { R } )$ ?
Hint: You may use question 3.