# Diagonalization of a covariance matrix

Consider $m=(m_i)$ a n-dimensional vector such that $m_i$ are i.i.d. standard Gaussian variables $N(0,1)$. Consider a T-dimensional vector $u=(u_t)$. We define $X_{i,t}=S(m_iu_t)$ where $S(.)$ is an odd function. The problem concerns the following matrix: $$R = X.X'$$ Do you have any idea about how I can find the eigenvalues/eigenvectors of $R$ ? It is very easy when $S(x)=x$, but I don't know how to find good approximations of the eigenvalues/eigenvectors of $R$ when $S$ is non linear.

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