The derivative of an orthogonal matrix of eigenvectors of a symmetric matrix

The authors supply the derivative of an orthogonal matrix of eigenvectors of a real symmetric matrix. To illustrate the applicability of their result they consider a real symmetric random matrix for which a more or less standard convergence in distribution is assumed to hold. The well-known delta method is then used to get the asymptotic distribution of the orthogonal eigenmatrix of the random matrix.