  The Mathieu equation occurs naturally in the description of non linear
vibrations or by considering the propagation of a wave in an infinite medium
with time-periodic refractive index. It is known to lead to parametric
instability since it supports unstable solutions in some regions of the
parameter space. However, even in the stable region the matrix that propagates
the initial conditions forward in time is non-normal and therefore it can
result in transient amplification. By optimizing over initial conditions as
well as initial time we show that significant transient amplifications can be
obtained, going beyond the one simply stemming from adiabatic invariance.
Moreover, we explore the monodromy matrix in more depth, by studying its
$\epsilon$-pseudospectra and Petermann factors, demonstrating that is the
degree of non-normality of this matrix that determines the global amplifying
features.
