A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincaré operator is used to project the eigenvalue problem on the domain
Ω
\Omega
onto the nonlinear eigenvalue subproblem on
Γ
\Gamma
, which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on
Γ
\Gamma
. Four different strategies are proposed to build the hierarchical subspace
U
k
+
1
U_{k+1}
over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace
U
k
+
1
U_{k+1}
. The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be
γ
=
1
−
c
0
T
h
,
H
−
1
\gamma =1-c_{0}T_{h,H}^{-1}
, where
c
0
c_{0}
is a constant independent of the fine mesh size
h
h
, the coarse mesh size
H
H
and jumps of the coefficients; whereas
T
h
,
H
T_{h,H}
is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.