  The scattering matrix for the full-line matrix Schr\"odinger equation is
analyzed when the corresponding matrix-valued potential is selfadjoint,
integrable, and has a finite first moment. The matrix-valued potential is
decomposed into a finite number of fragments, and a factorization formula is
presented expressing the matrix-valued scattering coefficients in terms of the
matrix-valued scattering coefficients for the fragments. Using the
factorization formula, some explicit examples are provided illustrating that in
general the left and right matrix-valued transmission coefficients are unequal.
A unitary transformation is established between the full-line matrix
Schr\"odinger operator and the half-line matrix Schr\"odinger operator with a
particular selfadjoint boundary condition and by relating the full-line and
half-line potentials appropriately. Using that unitary transformation, the
relations are established between the full-line and the half-line quantities
such as the Jost solutions, the physical solutions, and the scattering
matrices. Exploiting the connection between the corresponding full-line and
half-line scattering matrices, Levinson's theorem on the full line is proved
and is related to Levinson's theorem on the half line.
