It is well known that the sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a difficult problem. The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can be computed for a specific eigenvector basis, the so‐called adjacent eigenvector basis. These adjacent eigenvectors depend on individual variables, which makes the eigenvector derivative calculation elaborate and expensive from a computational perspective. This research presents a method that avoids passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. As our method fits into the adjoint sensitivity analysis , it is efficient for computing the complete Jacobian matrix because the adjoint variables are independent of each variable. Thus our method clarifies and unifies existing theories on eigenvector sensitivity analysis. Moreover, it provides a highly efficient computational method with a significant saving of the computational cost. Additional benefits of our approach are that one does not have to solve a deficient linear system and that the method is independent of the existence of repeated eigenvalue derivatives of the multiple eigenvalues. Our method covers the case of eigenvectors associated to a single eigenvalue. Some examples are provided to validate the present approach.

中文翻译：

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一种高效的重复特征值不经过相邻特征向量的特征向量敏感性分析的通用方法

众所周知，对应于多个特征值的特征向量的灵敏度分析是一个难题。主要困难在于，对于给定的多个特征值，可以针对特定特征向量计算特征向量导数，即所谓的相邻特征向量。特征向量的基础。这些相邻的特征向量取决于各个变量，从计算的角度来看，这使得特征向量导数计算复杂而昂贵。这项研究提出了一种在计算任何指定特征向量基的偏导数时避免通过相邻特征向量的方法。由于我们的方法适合于伴随灵敏度分析，因此它是计算完整的Jacobian矩阵的有效方法，因为伴随变量独立于每个变量。因此，我们的方法澄清和统一了本征矢量灵敏度分析的现有理论。此外，它提供了一种高效的计算方法，并且大大节省了计算成本。我们的方法的其他好处是不必解决线性系统的缺陷，并且该方法与多个特征值的重复特征值导数的存在无关。我们的方法涵盖了与单个特征值关联的特征向量的情况。提供了一些示例来验证本方法。