We now assume that $S _ { + } = S _ { - }$ and we assume that $\left( x _ { + } \mid q \right) = 0$. We denote by $z = \binom { x _ { + } } { 0 }$, $R ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , R ^ { - } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & - 1 \end{array} \right)$. (a) Show that $O z = R^{+} z = R^{-} z$. (b) We now write $$O = \left( \begin{array} { c c }
\alpha ^ { \prime } & { } ^ { t } q ^ { \prime } \\
r ^ { \prime } & P ^ { \prime }
\end{array} \right)$$ where $P ^ { \prime } \in M _ { n - 1 } ( \mathbb { R } )$. Construct then $z ^ { \prime } = \binom { \eta ^ { \prime } } { x ^ { \prime } } \in \mathbb { R } ^ { n }$ with $x ^ { \prime } \in \mathbb { R } ^ { n - 1 }$ strictly positive and $\eta ^ { \prime } \geq 0$ such that there exists a sign diagonal matrix $R ^ { \prime }$ satisfying $O z ^ { \prime } = R ^ { \prime } z ^ { \prime }$. (c) In the case where $\eta ^ { \prime } = 0$, and using question 1(c), show that there exists a sign diagonal matrix $S$ such that $O \left( z + z ^ { \prime } \right) = S \left( z + z ^ { \prime } \right)$ and conclude.
We now assume that $S _ { + } = S _ { - }$ and we assume that $\left( x _ { + } \mid q \right) = 0$. We denote by $z = \binom { x _ { + } } { 0 }$, $R ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , R ^ { - } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & - 1 \end{array} \right)$.\\
(a) Show that $O z = R^{+} z = R^{-} z$.\\
(b) We now write
$$O = \left( \begin{array} { c c }
\alpha ^ { \prime } & { } ^ { t } q ^ { \prime } \\
r ^ { \prime } & P ^ { \prime }
\end{array} \right)$$
where $P ^ { \prime } \in M _ { n - 1 } ( \mathbb { R } )$. Construct then $z ^ { \prime } = \binom { \eta ^ { \prime } } { x ^ { \prime } } \in \mathbb { R } ^ { n }$ with $x ^ { \prime } \in \mathbb { R } ^ { n - 1 }$ strictly positive and $\eta ^ { \prime } \geq 0$ such that there exists a sign diagonal matrix $R ^ { \prime }$ satisfying $O z ^ { \prime } = R ^ { \prime } z ^ { \prime }$.\\
(c) In the case where $\eta ^ { \prime } = 0$, and using question 1(c), show that there exists a sign diagonal matrix $S$ such that $O \left( z + z ^ { \prime } \right) = S \left( z + z ^ { \prime } \right)$ and conclude.