If $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5$ are the roots of the equation
$$ x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 = 0, $$
then find the value of
$$ \frac{1}{\alpha_1^2} + \frac{1}{\alpha_2^2} + \frac{1}{\alpha_3^2} + \frac{1}{\alpha_4^2} + \frac{1}{\alpha_5^2}. $$