## 1 Introduction  
In structural vibration modal analysis, the eigenvalue (square of frequency) and eigenvector (mode shape) are the two important parameters and their sensitivities are often used in many engineering problems such as structural vibration control [1; 2; 3; 4; 5; 6], optimization design [7; 8; 9], model updating [10; 11; 12; 13; 14] and damage identification [15; 16; 17]. From the current research literature, the calculation formula of eigenvalue sensitivity is simple, but the calculating formula for eigenvector sensitivity is complicated. At present, there are two kinds of methods to calculate the sensitivities of eigenvectors: exact methods and approximate methods. The classical exact methods are modal superposition method [18] and [PERSON]'s method [19]. The basic principle of mode superposition method is to express the eigenvector sensitivity as a linear combination of all modes. Thus modal superposition method has a great limitation in application because only partial modes are available for large-scale structures. [PERSON] et al. [20] studied the improved modal superposition method by approximating the eigenvector derivatives using only the available lower modes. [PERSON] and [PERSON] [21; 22] proposed an iterated modal superposition method to further improve numerical accuracy. Their method was only efficient for the case of limited numbers of design variables. [PERSON] [23] improved the computational efficiency of modal superposition method by using fixed basis model reduction technique. The modal superposition method has also been developed to treat with the damping systems [24; 25], and complex-valued systems [26; 27] by some researchers. In general, the calculation formulas of modal superposition method and its improved methods are complicated, especially for the higher-order sensitivity analysis. Compared with modal superposition method, [PERSON]'s method and its extended methods have an obvious advantage that only the eigenvector of interest is required. [PERSON] et al. [28; 29] improved the computational efficiency of [PERSON]'s method by using inverse iteration and model reduction techniques. [PERSON] and [PERSON] [30] used [PERSON]'s method to calculate the derivatives of the eigenvalues and eigenvectors for nonviscously damped systems. [PERSON] et al. [31] presented an improved [PERSON]'s Method for computing eigenvector derivatives with distinct and repeated eigenvalues. The improvement of their method is mainly to reduce the condition number of coefficient matrix. [PERSON] et al. [32] extended [PERSON]'s approach to the computation of the second-order eigenpair derivatives for symmetric and asymmetric damped systems with distinct eigenvalues. Using generalized inverse, [PERSON] and [PERSON] [33,34] further extended [PERSON]'s method to the calculation of eigensensitivities for symmetric and asymmetric damped systems with repeated eigenvalues. [PERSON] et al. [35] studied the computation method of eigenvector sensitivity in structural and topology optimization problems. They proposed a general framework for computing eigenvector sensitivity whenever tracking specific mode shapes selected beforehand. [PERSON] and [PERSON] [36] studied the computation problem of eigenpair sensitivities for fractional vibration systems. Their work examined how eigenvalue and eigenvector derivatives of fractional systems can be derived when system matrices become functions of physical design parameters. Although [PERSON]'s method and the above extended methods do not need to use all eigenvectors, there is no unified sensitivity calculation formula for each eigenvector, and the operation steps are slightly cumbersome. In view of this, this work presents a new exact method for calculating the sensitivities of eigenvectors. The explicit expressions for the first-order and second-order sensitivities of eigenvectors are derived, and strict proof is given. These formulas are simple and convenient in programming. The proposed method is as powerful as [PERSON]'s method, but much more easy to use.