The derivative of an orthogonal matrix of eigenvectors of a symmetric matrix (Q1369317)

Let \(M\) be a real symmetric \(p\times p\) matrix with distinct eigenvalues \(\lambda_i\) and associated normalized eigenvectors \(w_i\), \(1\leq i\leq p\). There are real-valued functions \(\psi_i\) and vector-valued functions \(f_i\) defined for all matrices \(Z\) in some neighborhood \({\mathcal N} (M) \subseteq \mathbb{R}^{p \times p}\) of \(M\), such that \(\psi_i(M)= \lambda_i\) and \(f_i(M) =w_i\), \(Zf_i= \psi_if_i\), \(f_i'f_j= \delta_{ij}\). If \(F=(f_i, \dots, f_p)\), a formula is given for a certain derivative of \(F\) and then applied to get the asymptotic distribution of the orthogonal eigenmatrix of a random matrix. This extends material found by \textit{J. R. Magnus} and \textit{H. Neudecker} [Matrix differential calculus with applications in statistics and econometrics, Wiley (1988; Zbl 0651.15001)].