Consider the function
$g(a) = \frac{1}{2}a^TQa-b^Ta$
Suppose we have the number condition of Q which is defined as $τ=\frac{v_n}{v_1}$ where $v_n$ is the largest eigenvalue and $v_1$ is the smallest eigenvalue. How can we show that
$||a_{k+1}-a_*|| <= \sqrt{τ} (\frac{τ-1}{τ+1})^k ||a_0-a_*||$
It is given that
$||a_{k+1}-a_*||<= (\frac{τ_n-τ_1}{τ_n+τ_1})^{2k} (||a_0-a_*||)$
I'm confused about how to derive something like this, especially when it involves inequality. Any help would be greatly appreciated. Thank you.
