  We study the breaking of time-reversal invariance (TRI) by the application of
a magnetic field in the quantum kicked rotor (QKR), using Izrailev's
finite-dimensional model. There is a continuous crossover from TRI to
time-reversal non-invariance (TRNI) in the spectral and eigenvector
fluctuations of the QKR. We show that the properties of this TRI $\rightarrow$
TRNI transition depend on $\alpha^2/N$, where $\alpha$ is the chaos parameter
of the QKR and $N$ is the dimensionality of the evolution operator matrix. For
$\alpha^2/N \gtrsim N$, the transition coincides with that in random matrix
theory. For $\alpha^2/N < N$, the transition shows a marked deviation from
random matrix theory. Further, the speed of this transition as a function of
the magnetic field is significantly enhanced as $\alpha^2/N$ decreases.
