  We present a first-order aggregation model on the space of complex matrices
which can be derived from the Lohe tensor model on the space of tensors with
the same rank and size. We call such matrix-valued aggregation model as "the
generalized Lohe matrix model". For the proposed matrix model with two cubic
coupling terms, we study several structural properties such as the conservation
laws, solution splitting property. In particular, for the case of only one
coupling, we reformulate the reduced Lohe matrix model into the Lohe matrix
model with a diagonal frustration, and provide several sufficient frameworks
leading to the complete and practical aggregations. For the estimates of
collective dynamics, we use a nonlinear functional approach using an ensemble
diameter which measures the degree of aggregation.
