  Random contractions (sub-unitary random matrices) appear naturally when
considering quantized chaotic maps within a general theory of open linear
stationary systems with discrete time. We analyze statistical properties of
complex eigenvalues of generic $N\times N$ random matrices $\hat{A}$ of such a
type, corresponding to systems with broken time-reversal invariance. Deviations
from unitarity are characterized by rank $M\le N$ and a set of eigenvalues
$0<T_i\le 1, i=1,...,M$ of the matrix $\hat{T}=\hat{{\bf
1}}-\hat{A}^{\dagger}\hat{A}$. We solve the problem completely by deriving the
joint probability density of $N$ complex eigenvalues and calculating all $n-$
point correlation functions. In the limit $N>>M,n$ the correlation functions
acquire the universal form found earlier for weakly non-Hermitian random
matrices.
