  In the previous paper (hep-th/0402010) we proposed a matrix configuration for
a non-commutative S^4 (NC4S) and constructed a non-commutative (star) product
for field theories on NC4S. In the present paper we will show that any matrix
can be expanded in terms of the matrix configuration representing NC4S just
like any matrix can be expanded into symmetrized products of the matrix
configuration for non-commutative S^2. Then a scalar field theory on NC4S is
constructed. Our matrix configuration describes two S^4's joined at the circle
and the Matrix theory action contains a projection matrix inside the trace to
restrict the space of matrices to that for one S^4.
