  In this paper we give a profound insight into the computation capability of
delay-based reservoir computing via an eigenvalue analysis. We concentrate on
the task-independent memory capacity to quantify the reservoir performance and
compare these with the eigenvalue spectrum of the dynamical system. We show
that these two quantities are deeply connected, and thus the reservoir
computing performance is predictable by analyzing the small signal response of
the reservoir. Our results suggest that any dynamical system used as a
reservoir can be analyzed in this way. We apply our method exemplarily to a
photonic laser system with feedback and compare the numerically computed recall
capabilities with the eigenvalue spectrum. Optimal performance is found for a
system with the eigenvalues having real parts close to zero and off-resonant
imaginary parts.
