We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ Using the induction hypothesis for $Q _ { + }$ (resp. for $Q _ { - }$), we denote by $x _ { + } > 0$ (resp. $x _ { - } > 0$) a vector of $\mathbb { R } ^ { n - 1 }$ and $S _ { + }$ (resp. $S _ { - }$) the sign diagonal matrix, such that $$Q _ { + } x _ { + } = S _ { + } x _ { + } , \quad \text { resp. } \quad Q _ { - } x _ { - } = S _ { - } x _ { - }$$
Show that $$\left( S _ { + } x _ { + } \mid S _ { - } x _ { - } \right) = \left( x _ { + } \mid x _ { - } \right) - \frac { 2 } { 1 - \alpha ^ { 2 } } \left( x _ { + } \mid q \right) \left( x _ { - } \mid q \right)$$