  We consider a class of eigenvector-dependent nonlinear eigenvalue problems
(NEPv) without the unitary invariance property. Those NEPv commonly arise as
the first-order optimality conditions of a particular type of optimization
problems over the Stiefel manifold, and previously, special cases have been
studied in the literature. Two necessary conditions, a definiteness condition
and a rank-preserving condition, on an eigenbasis matrix of the NEPv that is a
global optimizer of the associated optimization problem are revealed, where the
definiteness condition has been known for the special cases previously
investigated. We show that, locally close to the eigenbasis matrix satisfying
both necessary conditions, the NEPv can be reformulated as a unitarily
invariant NEPv, the so-called aligned NEPv, through a basis alignment operation
-- in other words, the NEPv is locally unitarily invariantizable. Numerically,
the NEPv is naturally solved by an SCF-type iteration. By exploiting the
differentiability of the coefficient matrix of the aligned NEPv, we establish a
closed-form local convergence rate for the SCF-type iteration and analyze its
level-shifted variant. Numerical experiments confirm our theoretical results.
