  Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but
their convergence theory remains sparse and complex. We consider the simplest
preconditioned eigensolver--the gradient iterative method with a fixed step
size--for symmetric generalized eigenvalue problems, where we use the gradient
of the Rayleigh quotient as an optimization direction. A sharp convergence rate
bound for this method has been obtained in 2001--2003. It still remains the
only known such bound for any of the methods in this class. While the bound is
short and simple, its proof is not. We extend the bound to Hermitian matrices
in the complex space and present a new self-contained and significantly shorter
proof using novel geometric ideas.
