任意非亏损矩阵特征灵敏度分析的模态展开法

Modal Expansion Method for Eigensensitivity Analysis of Arbitrary Non-Defective Matrices

把特征向量的各阶导数表示成所有模态的线性组合,并利用左模态与右模态间的双正交性,首先导出了任意非亏损矩阵的重特征值的一阶导数所满足的特征值问题,然后根据此特征值问题无、有重根的情况,再导出了异导重特征值和等导重特征值对应的可微特征向量、特征值和特征向量各阶导数的一般计算公式。算例显示了方法的正确性。

By expressing the eigenvector derivatives as linear combinations of all of the modes and using the bi-orthogonality of the left modes with the right modes, the eigenvalue problem for first-order derivatives of repeated eigenvalues was derived first and then, according to the eigenvalue problem without or with multiple roots, the general formulas for calculating the differentiable eigenvectors and any order of derivatives of eigenvalues and eigenvectors of an arbitrary non-defective matrix with distinct or repeated first-order eigenvalue derivatives were derived. Numerical example shows the correctness of the method.