I have been working this question for a while, and got stuck for 1 hour so I decided to ask for help. This is maily due to my rusty math skill working with matrix.
$J(x_t) = \frac{1}{2}(x_t-x^*)^TP(x_t-x^*)$
where $P\in \mathbb R^{n\times n}$, P is positive definite, $a, x_t, x^* \in \mathbb R^n$, with $x^*$ fixed.
$$x_{t+1} = x_t - \frac{g_t^Tg_t}{g_t^TPg_t} g_t, $$ where $g_t = Px_t+a$.
Show that $J(x_{t+1}) = \left(1-\frac{(g^T_tg_t)^2}{(g_t^TPg_t)(g_t^TP^{-1}g_t)}\right)J(x_t)$.
Hint: Consider using $(J(x_t) - J(x_{t+1})) / (J(x_t)$.
Attempt:
I try to basically substitute $x_{t+1}$ onto J, expand it , and got some dirty equations and unable to make it the form they requre. I try to do the reverse direction as well, but I am not able to work my way out.