Truncations of random unitary matrices

We analyse properties of non-Hermitian matrices of size M constructed as sq uare submatrices of unitary (orthogonal) random matrices of size N > M, dis tributed according to the Haar measure. In this way we define ensembles of random matrices and study the statistic;al properties of the spectrum locat ed inside the unit circle. In the limit of large matrices, this ensemble is characterized by the ratio M/N. For the truncated CUE we analytically deri ve the joint density of eigenvalues and all correlation functions. In the s trongly non-unitary case universal Ginibre behaviour is found. For N - M fi xed and N to infinity the universal resonance-width distribution with N - M open channels is recovered.