  A Borg-type uniqueness theorem for matrix-valued Schr\"odinger operators is
proved. More precisely, assuming a reflectionless potential matrix and spectrum
a half-line $[0,\infty)$, we derive triviality of the potential matrix. Our
approach is based on trace formulas and matrix-valued Herglotz representation
theorems. As a by-product of our techniques, we obtain an extension of Borg's
classical result from the class of periodic scalar potentials to the class of
reflectionless matrix-valued potentials.
