  The Jarzynski equality (JE), which relates works of non-equilibrium
trajectories to the free energy difference of the initial and final states of
the non-equilibrium process, provides an efficient way to calculate free
energies of systems in simulations and experiments. However, wider applications
of the JE are limited by the requirement that the initial distribution of
non-equilibrium trajectories must be equilibrium. Here we extend the JE to a
matrix form, the Jarzynski matrix equality (JME), which transforms the free
energies of metastable conformational regions in the initial system to that of
final one. Therefore, we can calculate the free energies from non-equilibrium
trajectories which started from an arbitrary initial distribution. We
demonstrate the application of the JME in toy models, Lennard-Jones fluids, and
polymer chain models, show its good efficiency in calculation of free energy
with a satisfactory accuracy. The JME extends applications of the
non-equilibrium method in estimate of free energy in complex system where the
initial global equilibrium is difficult to reach.
