## 3 The New Exact Method for Eigenvector Sensitivities  
In view of the shortcomings of existing methods, the paper proposes a new exact method for calculating the eigenvector sensitivity. The main formulas are derived as follows. Firstly, Equation (3) can be rewritten by adding a term of \(\lambda_{j}\phi_{j}\phi_{j}^{T}M\frac{\partial\phi_{j}}{\partial p_{i}}\) as  
\[(K-\lambda_{j}M+\lambda_{j}\phi_{j}\phi_{j}^{T}M)\frac{\partial\phi_{j}}{ \partial p_{i}}=(\frac{\partial\lambda_{j}}{\partial p_{i}}M+\lambda_{j}\frac {\partial M}{\partial p_{i}}-\frac{\partial K}{\partial p_{i}})\phi_{j}+ \lambda_{j}\phi_{j}\phi_{j}^{T}M\frac{\partial\phi_{j}}{\partial p_{i}} \tag{17}\]  
substituting Equation (6) into \(\lambda_{j}\phi_{j}\phi_{j}^{T}M\frac{\partial\phi_{j}}{\partial p_{i}}\) yields  
\[\lambda_{j}\phi_{j}\phi_{j}^{T}M\frac{\partial\phi_{j}}{\partial p_{i}}=\sum_ {r=1,r\
eq j}^{n}c_{jr}\lambda_{j}\phi_{j}(\phi_{j}^{T}M\phi_{r})-c_{ji}\lambda _{j}\phi_{j}(\phi_{j}^{T}M\phi_{j}) \tag{18}\]  
according to the orthogonally of eigenvectors, one has  
\[\phi_{j}^{T}M\phi_{r}=0,\ j\
eq r \tag{19}\]  
substituting Equations (2) and (19) into Equation (18), one can obtain  
\[\lambda_{j}\phi_{j}\phi_{j}^{T}M\frac{\partial\phi_{j}}{\partial p_{i}}=-c_{ji }\lambda_{j}\phi_{j} \tag{20}\]  
substituting Equation (20) into Equation (17) yields  
\[\Theta\cdot\frac{\partial\phi_{j}}{\partial p_{i}}=\Pi\cdot\phi_{j} \tag{21}\]  
where  
\[\Theta=K-\lambda_{j}M+\lambda_{j}\phi_{j}\phi_{j}^{T}M \tag{22}\]\[\Pi=\frac{\partial\lambda_{j}}{\partial p_{i}}M+\lambda_{j}\frac{\partial M}{ \partial p_{i}}-\frac{\partial K}{\partial p_{i}}-c_{jj}\lambda_{j}I \tag{23}\]  
If the matrix \(\Theta\) is full rank, the eigenvector sensitivity \(\frac{\partial\phi_{j}}{\partial p_{i}}\) can be directly computed from Equation (21) as:  
\[\frac{\partial\phi_{j}}{\partial p_{i}}=\Theta^{-1}\Pi\phi_{j} \tag{24}\]  
The proposition that the matrix \(\Theta\) is full rank can be proved as follows. Proof. : To prove the above proposition, it is necessary to use for reference the related viewpoints and proof process of the Cayley-Hamilton theorem. Matrix diagonalization is one of the common techniques used in the proof of Cayley-Hamilton theorem. Therefore, in the next proof process, matrix diagonalization is also used as the main tool to simplify the matrix \(\Theta\). Letting  
\[\Phi=[\phi_{1},\phi_{2},\cdots,\phi_{n}] \tag{25}\]  
It is clear that the matrix \(\Phi\) consisting of all the eigenvectors is full rank. Then the original proposition \"\(\Theta\) is full rank\" can be equivalent to the new proposition \"\(\Phi^{T}\Theta\Phi\) is full rank\". Using Equations (22) and (25), \(\Phi^{T}\Theta\Phi\) can be expanded as  
\[\Phi^{T}\Theta\Phi=\Phi^{T}K\Phi-\lambda_{j}\Phi^{T}M\Phi+\lambda_{j}\Phi^{T }\phi_{j}\phi_{j}^{T}M\Phi \tag{26}\]  
where  
\[\Phi^{T}K\Phi=\left[ \begin{array}{cccc}\lambda_{1}&&&\\ &\lambda_{2}&&\\ &&\ddots&\\ &&&\lambda_{n}\end{array} \right] \tag{27}\]  
\[\lambda_{j}\Phi^{T}M\Phi=\lambda_{j}\left[ \begin{array}{cccc}1&&&\\ &1&&\\ &&\ddots&\\ &&&1\end{array} \right] \tag{28}\]  
\[\lambda_{j}\Phi^{T}\phi_{j}\phi_{j}^{T}M\Phi=\lambda_{j}\left[ \begin{array}[] {cccccc}0&\cdots&0&\phi_{1}^{T}\phi_{j}&0&\cdots&0\\ 0&\cdots&0&\phi_{2}^{T}\phi_{j}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&0&\phi_{1}^{T}\phi_{j}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&0&\phi_{n}^{T}\phi_{j}&0&\cdots&0\end{array} \right] \tag{29}\]  
Substituting Equations (27)-(29) into Equation (26) yields  
\[\Phi^{T}\Theta\Phi=\left[