  Given $n$ linearly independent pure states and their prior probabilities, we
study the problem of optimum unambiguous discrimination of these states. We
derive the properties of the optimum solution and the equations that must be
satisfied by the optimum measurement strategy which achieves the maximum
average success probability, and also give the detailed steps to obtain the
optimum solution and the optimum measurement strategy. The general method and
results we obtain are also illustrated both numerically and geometrically. We
derive a simple analytical formula of the maximum average success probability
of unambiguous discrimination for a given set of pure states, and it can be
used to simplify the calculation of the optimum solution in some situations. We
also obtain the analytical solution of a generalized equal-probability
measurement problem using the equations we introduce. Finally, as another
application of our result, we study the optimum unambiguous discrimination
problem of three linearly independent pure states in details and obtain
analytical solutions for some special cases.
