  The Jacobi-Davidson method is one of the most popular approaches for
iteratively computing a few eigenvalues and their associated eigenvectors of a
large matrix. The key of this method is to expand the search subspace via
solving the Jacobi-Davidson correction equation, whose coefficient matrix is
singular. It is believed long by scholars that the Jacobi-Davidson correction
equation is a consistent linear system. In this work, we point out that the
correction equation may have a unique solution or have no solution at all, and
we derive a computable necessary and sufficient condition for cheaply judging
the existence and uniqueness of solution of the correction equation.
Furthermore, we consider the difficulty of stagnation that bothers the
Jacobi-Davidson method, and verify that if the Jacobi-Davidson method
stagnates, then the corresponding Ritz value is a defective eigenvalue of the
projection matrix. We provide a computable necessary and sufficient condition
for expanding the search subspace successfully. The properties of the
Jacobi-Davidson method with preconditioning and some alternative
Jacobi-Davidson correction equations are also discussed.
