An alternative formulation is proposed for the calculation of the non-adiab
atic non-Abelian connection and phase factor by making use of the invariant
theory and invariant-related unitary transformation method. In order to il
lustrate the formulation, we study the non-adiabatic non-Abelian phase asso
ciated with the motion of a charged particle in a time-dependent magnetic f
ield (the time-dependent Landau level problem).