  This paper is concerned with the extraction of the smallest eigenvalue and
the corresponding eigenvector of a symmetric positive definite matrix pencil.
We reveal implicit convexity of the eigenvalue problem in Euclidean space. A
provable accelerated eigensolver based on preconditioning and implicit
convexity (EPIC) is proposed. Theoretical analysis shows the acceleration of
EPIC with the rate of convergence resembling the expected rate of convergence
of the well-known locally optimal preconditioned conjugate gradient (LOPCG). A
complete proof of the expected rate of convergence of LOPCG is elusive so far.
Numerical results confirm our theoretical findings of EPIC.
