## 3 The New Exact Method for Eigenvector Sensitivities
\begin{array}{cccccc}\lambda_{1}-\lambda_{j}& \cdots&0&\lambda_{j}\phi_{1}^{T}\phi_{j}&0&\cdots&0\\ 0&\ddots&0&\lambda_{j}\phi_{2}^{T}\phi_{j}&0&\cdots&0\\ \vdots&\vdots&\lambda_{j-1}-\lambda_{j}&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&0&\lambda_{j}\phi_{j}^{T}\phi_{j}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\lambda_{j+1}-\lambda_{j}&\vdots&\vdots\\ 0&\cdots&0&\lambda_{j}\phi_{n+1}^{T}\phi_{j}&\vdots&\ddots&0\\ 0&\cdots&0&\lambda_{j}\phi_{n}^{T}\phi_{j}&0&\cdots&\lambda_{n}-\lambda_{j} \end{array} \right] \tag{30}\]Now the proposition \(\Phi^{T}\Theta\Phi\) is full rank can be equivalent to the new proposition \"the Equation \(\Phi^{T}\Theta\Phi\cdot\left\{ \begin{array}{c}\delta_{1}\\ \vdots\\ \delta_{n}\end{array} \right\}=\left\{ \begin{array}{c}0\\ \vdots\\ 0\end{array} \right\}\) only has zero solution\". Using Equation (30), \(\Phi^{T}\Theta\Phi\cdot\left\{ \begin{array}{c}\delta_{1}\\ \vdots\\ \delta_{n}\end{array} \right\}=\left\{ \begin{array}{c}0\\ \vdots\\ 0\end{array} \right\}\) can be simplified as  
\[\left\{ \begin{array}{c}\delta_{1}(\lambda_{1}-\lambda_{j})+\delta_{j}\lambda _{j}\phi_{1}^{T}\phi_{j}\\ \vdots\\ \delta_{j-1}(\lambda_{j-1}-\lambda_{j})+\delta_{j}\lambda_{j}\phi_{j-1}^{T} \phi_{j}\\ \delta_{j+1}(\lambda_{j+1}-\lambda_{j})+\delta_{j}\lambda_{j}\phi_{j+1}^{T} \phi_{j}\\ \vdots\\ \delta_{n}(\lambda_{n}-\lambda_{j})+\delta_{j}\lambda_{j}\phi_{n}^{T}\phi_{j} \end{array} \right\}=\left\{ \begin{array}{c}0\\ \vdots\\ 0\\ 0\\ \vdots\\ 0\end{array} \right\} \tag{31}\]  
from the \(j\)th row of Equation (31), one has  
\[\delta_{j}=0 \tag{32}\]  
substituting Equation (32) into Equation (31), one can obtain  
\[\delta_{1}=\delta_{2}=\cdots=\delta_{n}=0 \tag{33}\]  
Equation (33) means that the equation \(\Phi^{T}\Theta\Phi\cdot\left\{ \begin{array}{c}\delta_{1}\\ \vdots\\ \delta_{n}\end{array} \right\}=\left\{ \begin{array}{c}0\\ \vdots\\ 0\end{array} \right\}\) only has zero solution. Thus the matrix \(\Phi^{T}\Theta\Phi\) is full rank. It proved that the matrix \(\Theta\) is full rank. End (End of the proof). \(\Box\)  
It must be pointed out that only systems with distinct eigenvalues are considered in this paper. If there exists two different eigenvectors with the same eigenvalue, the matrix presented in Equation (30) is not at full rank and therefore the above thesis will be invalid. The problem is known and presented in the published literature. The existing methods to address this problem can be found in references [23; 31; 33; 34; 35]. How to extend the proposed method to the systems with repeated eigenvalues will be further studied in the future. Equation (24) is the proposed new formula for calculating the first-order sensitivity of the eigenvector. Obviously, this formula has the advantage of requiring only the knowledge of the eigenvector to be differentiated, and there is no need for multi-step operations like [PERSON]'s method. The formula is concise and can be applied to any eigenvector. Another advantage of the proposed method is that it can be easily extended to the calculations of the second- and higher order sensitivities of eigenvectors. Without loss of generality, the calculation formulas of the second-order sensitivity are derived as follows.