We consider the Schr\"odinger operator on the real line with a 2x2 matrix
valued 1-periodic potential. The spectrum of this operator is absolutely
continuous and consists of intervals separated by gaps. We define a Lyapunov
function which is analytic on a two sheeted Riemann surface. On each sheet, the
Lyapunov function has the same properties as in the scalar case, but it has
branch points, which we call resonances. We prove the existence of real as well
as non-real resonances for specific potentials. We determine the asymptotics of
the periodic and anti-periodic spectrum and of the resonances at high energy.
We show that there exist two type of gaps: 1) stable gaps, where the endpoints
are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where
the endpoints are resonances (i.e., real branch points of the Lyapunov
function). We also show that periodic and anti-periodic spectrum together
determine the spectrum of the matrix Hill operator.