We consider quadratic forms of deterministic matrices $A$ evaluated at the
random eigenvectors of a large $N \times N$ GOE or GUE matrix, or equivalently
evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We
prove that, as long as the deterministic matrix has rank much smaller than
$\sqrt{N}$, the distributions of the extrema of these quadratic forms are
asymptotically the same as if the eigenvectors were independent Gaussians. This
reduces the problem to Gaussian computations, which we carry out in several
cases to illustrate our result, finding Gumbel or Weibull limiting
distributions depending on the signature of $A$. Our result also naturally
applies to the eigenvectors of any invariant ensemble.
