  We study the generalizations of the known equivalent reformulations of
condition moderate growth from the single weight sequence to the weight matrix
setting. This condition, also known in the literature under the name stability
under ultradifferentiable operators, plays a significant role in the theory of
ultradifferentiable (and ultraholomorophic) function classes defined in terms
of weight sequences and its generalization becomes relevant when dealing with
classes defined by weight matrices. In the matrix setting, we prove that the
different mixed conditions are in general not equivalently satisfied anymore
and we focus on weight matrices associated with (associated) weight functions.
