  We study joint eigenvector distributions for large symmetric matrices in the
presence of weak noise. Our main result asserts that every submatrix in the
orthogonal matrix of eigenvectors converges to a multidimensional Gaussian
distribution. The proof involves analyzing the stochastic eigenstate equation
(SEE) which describes the Lie group valued flow of eigenvectors induced by
matrix valued Brownian motion. We consider the associated colored eigenvector
moment flow defining an SDE on a particle configuration space. This flow
extends the eigenvector moment flow first introduced in Bourgade and Yau (2017)
to the multicolor setting. However, it is no longer driven by an underlying
Markov process on configuration space due to the lack of positivity in the
semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged
decay and contractive properties. This allows us to establish optimal time of
relaxation to equilibrium for the colored eigenvector moment flow and prove
joint asymptotic normality for eigenvectors. Applications in random matrix
theory include the explicit computations of joint eigenvector distributions for
general Wigner type matrices and sparse graph models when corresponding
eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector
distributions for L\'evy matrices when the eigenvectors correspond to small
energy levels.
