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.

翻译：本文研究一类不具有酉不变性的特征向量依赖非线性特征值问题（NEPv）。这类NEPv通常作为Stiefel流形上特定类型优化问题的一阶最优性条件出现，此前文献中已研究过其特例。揭示了作为相关优化问题全局最优解的NEPv特征基矩阵的两个必要条件——正定性条件与秩保持条件，其中正定性条件在先前研究的特例中已知。我们证明，在同时满足两个必要条件的特征基矩阵局部邻域内，通过基对齐操作可将该NEPv重构为酉不变NEPv（即所谓的对齐NEPv）——换言之，该NEPv是局部酉不变可化的。数值计算中，该NEPv自然通过SCF型迭代求解。通过利用对齐NEPv系数矩阵的可微性，我们建立了SCF型迭代的闭式局部收敛率，并分析了其平移变体。数值实验验证了理论结果。