  Matrix evolution equations occur in many applications, such as dynamical
Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic
control, machine learning or data assimilation. In many cases, their tightest
stability condition is coming from a linear term. Exponential time differencing
(ETD) is known to produce highly stable numerical schemes by treating the
linear term in an exact fashion. In particular, for stiff problems, ETD methods
are a method of choice. We propose an extension of the class of ETD algorithms
to matrix-valued dynamical equations. This allows us to produce highly
efficient and stable integration schemes. We show their efficiency and
applicability for a variety of real-world problems, from geophysical
applications to dynamical problems in machine learning.
