  The Moody-Shapere-Wilczek's adiabatic effective Hamiltonian and Lagrangian
method is developed further into the matrix effective Hamiltonian (MEH) and
Lagrangian (MEL) approach to a parameter-dependent quantum system. The
matrix-operator approach formulated in the product integral (PI) provides not
only a method to find the wave function efficiently in the MEH approach but
also higher order corrections to the effective action systematically in the MEL
approach, a la the Magnus expansion and the Kubo cumulant expansion. A coupled
quantum system of a light particle of a harmonic oscillator is worked out, and
as a by-product, a new kind of gauge potential (Berry's connection) is found
even for nondegenerate cases (real eigenfunctions). Moreover, in the PI
formulation the holonomy of the induced gauge potential is related to
Schlesinger's exact formula for the gauge field tensor. A superadiabatic
expansion is also constructed, and a generalized Dykhne formula, depending on
the contour integrals of the homotopy class of complex degenerate points, is
rephrased in the PI formulation.
