Distribution of eigenvalues and eigenvectors of orthogonal random matrices (Q1074214)

To find the distribution of eigenvalues and eigenvectors of random matrices is a central problem of the spectral theory of random matrices. In his previous works the author has found the distribution of eigenvalues and eigenvectors of symmetric, Hermitian, skew-symmetric, non-symmetric, complex, Gaussian and unitary random matrices. In this paper the author deals with orthogonal random matrices. As known, with probability one each orthogonal real random matrix \(H_ n\) can be expressed in the form \[ H_ n=\theta_ ndiag\{\left( \begin{matrix} \cos \lambda_ 1\quad \sin \lambda_ 1\\ -\sin \lambda_ 1\quad \cos \lambda_ 1\end{matrix} \right),...,\left( \begin{matrix} \cos \lambda_ q\quad \sin \lambda_ q\\ -\sin \lambda_ q\quad \cos \lambda_ q\end{matrix}\right),\quad +1,\quad -1\}\theta '_ n \] for even n and \[ H_ n=\theta_ ndiag\{\left( \begin{matrix} \cos \lambda_ 1\quad \sin \lambda_ 1\\ -\sin \lambda_ 1\quad \cos \lambda_ 1\end{matrix} \right),...,\left( \begin{matrix} \cos \lambda_ p\quad \sin \lambda_ p\\ - \sin \lambda_ p\quad \cos \lambda_ p\end{matrix} \right),\quad \xi \}\theta '_ n \] for odd n where \(\theta_ n\) is an orthogonal matrix. The author has given explicit formulas for finding the joint distribution of \(\theta_ n\) and \(\lambda_ k\) (they are too complicated to present them here!). The author's results are certainly very useful.