  The theory of inverse scattering is developed to study the initial-value
problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the
$2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at
infinity. In the direct problem, by introducing a suitable uniform
transformation we establish the proper complex $z$-plane in order to discuss
the Jost eigenfunctions, scattering matrix and their analyticity and symmetry
of the equation. Moreover the asymptotic behavior of the Jost functions and
scattering matrix needed in the inverse problem are analyzed via
Wentzel-Kramers-Brillouin expansion. In the inverse problem, the generalized
Riemann-Hilbert problem of the mmKdV equation is first established by using the
analyticity of the modified eigenfunctions and scattering coefficients. The
reconstruction formula of potential function with reflection-less case is
derived by solving this Riemann-Hilbert problem and using the scattering data.
In addition the dynamic behavior of the solutions for the focusing mmKdV
equation including one- and two- soliton solutions are presented in detail
under the the condition that the potential is scalar and the $2\times2$
symmetric matrix. Finally, we provide some detailed proofs and weak version of
trace formulas to show that the asymptotic phase of the potential and the
scattering data.
