A precise answer exists for the Gaussian Orthogonal Ensemble (all variances the same): then the eigenvectors are the columns of an orthogonal matrix which is uniformly distributed with the Haar measure; they are therefore not independent --- they cannot be because they must be orthogonal to one another.

In the limit $n\rightarrow\infty$ of large $n\times n$ matrices $M$, any finite subset of elements of the eigenvectors does become statistically independent with a Gaussian distribution (mean zero, variance $1/n$). This "central limit" result does not require that the elements of $M$ have identical distributions.

A precise answer exists for the Gaussian Orthogonal Ensemble (all variances the same): then the eigenvectors are the columns of an orthogonal matrix which is uniformly distributed with the Haar measure; they are therefore not independent --- they cannot be because they must be orthogonal to one another.

In the limit $n\rightarrow\infty$ of large $n\times n$ matrices $M$, any finite subset of elements of the eigenvectors does become statistically independent [1] with a Gaussian distribution (mean zero, variance $1/n$). This "central limit" result does not require that the elements of $M$ have identical distributions.

[1] How Many Entries of a Typical Orthogonal Matrix Can Be Approximated by Independent Normals? (2006).

A precise answer exists for the Gaussian Orthogonal Ensemble (all variances the same): then the eigenvectors are the columns of an orthogonal matrix which is uniformly distributed with the Haar measure; they are therefore not independent, they cannot be because they must be orthogonal.

In the limit $n\rightarrow\infty$ of large $n\times n$ matrices, any finite subset of elements of the eigenvectors does become statistically independent with a Gaussian distribution (mean zero, variance $1/n$).

A precise answer exists for the Gaussian Orthogonal Ensemble (all variances the same): then the eigenvectors are the columns of an orthogonal matrix which is uniformly distributed with the Haar measure; they are therefore not independent --- they cannot be because they must be orthogonal to one another.

In the limit $n\rightarrow\infty$ of large $n\times n$ matrices $M$, any finite subset of elements of the eigenvectors does become statistically independent with a Gaussian distribution (mean zero, variance $1/n$). This "central limit" result does not require that the elements of $M$ have identical distributions.