I can understand the half-length in the direction $\textbf{e}_1$ is $c/\sqrt{\lambda _1}$ , given $c^2 = \lambda_1 y_1^2 + \lambda_2 y_2^2 $, where $y_1 = x'e_1$ and $y_2=x'e_2$.
But I don't understand why we can safely claim that the direction of hyperellipsoid is just eigenvectors $e_1, e_2$?