A new formulation is proposed for the investigation of the non-adiabatic no
n-Abelian connection and phase factor by making use of the invariant theory
and invariant-related unitary transformation method. In order to illustrat
e the formulation, we calculate the non-adiabatic non-Abelian connection an
d phase factor associated with the motion of a charged particle in a time-d
ependent magnetic field (the time-dependent Landau level problem).