  In M(atrix) theory, there exist membranes and longitudinal 5-branes
(L5-branes) as extended objects. Transverse components of these brane solutions
are known to be described by fuzzy CP^k (k=1,2), where k=1 and k=2 correspond
to spherical membranes and L5-branes of CP^2 \times S^1 world-volume geometry,
respectively. In addition to these solutions, we here show the existence of
L7-branes of CP^3 \times S^1 geometry, introducing extra potentials to the
M(atrix) theory Lagrangian. As in the cases of k=1,2, the L7-branes
(corresponding to k=3) also break the supersymmetries of M(atrix) theory. The
extra potentials are introduced such that the energy of a static L7-brane
solution becomes finite in the large N limit where N represents the matrix
dimension of fuzzy CP^3. As a consequence, fluctuations from the L7-branes are
suppressed, which effectively describes compactification of M(atrix) theory
down to 7 dimensions. We show that one of the extra potentials can be
considered as a matrix-valued 7-form. The presence of the 7-form in turn
supports a possibility of Freund-Rubin type compactification. This suggests
that our modification of M(atrix) theory can also lead to a physically
interesting matrix model in four dimensions. In hope of such a possibility, we
further consider compactification of M(atrix) theory down to fuzzy S^4 which
can be defined in terms of fuzzy CP^3. Along the way, we also find a new
L5-brane solution to M(atrix) theory which has purely spherical geometry in the
transverse directions.
