MPEJ Volume 13, No. 5, 40 pp. Received: Mar 1, 2007. Revised: Aug 14, 2007. Accepted: Sep 20, 2007. H.Schulz-Baldes Rotation numbers for Jacobi matrices with matrix entries ABSTRACT: A Jacobi matrix with matrix entries is a selfadjoint block tridiagonal matrix with positive definite blocks on the off-diagonals. A rotation number calculation for its eigenvalues is presented. This is a matricial generalization of the oscillation theorem for the discrete analogues of Sturm-Liouville operators. The three universality classes of time reversal invariance are dealt with by implementing the corresponding symmetries. For Jacobi matrices with random matrix entries, this leads to a formula for the integrated density of states which can be calculated perturbatively in the coupling constant of the randomness with an optimal control on the error terms. http://www.maia.ub.es/mpej/Vol/13/5.ps http://www.maia.ub.es/mpej/Vol/13/5.pdf http://www.ma.utexas.edu/mpej/Vol/13/5.ps http://www.ma.utexas.edu/mpej/Vol/13/5.pdf http://mpej.unige.ch/mpej/Vol/13/5.ps http://mpej.unige.ch/mpej/Vol/13/5.pdf