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The scattering matrix for the full-line matrix Schr\"odinger equation isanalyzed when the corresponding matrix-valued potential is selfadjoint,integrable, and has a finite first moment. The matrix-valued potential isdecomposed into a finite number of fragments, and a factorization formula ispresented expressing the matrix-valued scattering coefficients in terms of thematrix-valued scattering coefficients for the fragments. Using thefactorization formula, some explicit examples are provided illustrating that ingeneral the left and right matrix-valued transmission coefficients are unequal.A unitary transformation is established between the full-line matrixSchr\"odinger operator and the half-line matrix Schr\"odinger operator with aparticular selfadjoint boundary condition and by relating the full-line andhalf-line potentials appropriately. Using that unitary transformation, therelations are established between the full-line and the half-line quantitiessuch as the Jost solutions, the physical solutions, and the scatteringmatrices. Exploiting the connection between the corresponding full-line andhalf-line scattering matrices, Levinson's theorem on the full line is provedand is related to Levinson's theorem on the half line.

Factorization for the full-line matrix Schr\"odinger equation and a unitary transformation to the half-line scattering