## 3 The New Exact Method for Eigenvector Sensitivities
Differentiating Equation (3) a second time with respect to a (possibly) different parameter \(p_{s}\), one has  
\[(K-\lambda_{j}M)\frac{\partial^{2}\phi_{j}}{\partial p_{i}\partial p_{s}}= \frac{\partial(\lambda_{j}M-K)}{\partial p_{s}}\frac{\partial\phi_{j}}{ \partial p_{i}}+\frac{\partial(\lambda_{j}M-K)}{\partial p_{i}}\frac{\partial \phi_{j}}{\partial p_{s}}+\frac{\partial^{2}(\lambda_{j}M-K)}{\partial p_{i} \partial p_{s}}\phi_{j} \tag{34}\]  
Multiplying Equation (34) by \(\phi_{j}^{T}\) and using Equations (2) and (4), the second sensitivity of \(\lambda_{j}\) can be obtained as:  
\[\frac{\partial^{2}\lambda_{j}}{\partial p_{i}\partial p_{s}}=\phi_{j}^{T}[ \frac{\partial^{2}K}{\partial p_{i}\partial p_{s}}-\lambda_{j}\frac{\partial^ {2}M}{\partial p_{i}\partial p_{s}}-\frac{\partial\lambda_{j}}{\partial p_{i }}\frac{\partial M}{\partial p_{s}}-\frac{\partial\lambda_{j}}{\partial p_{i }}\frac{\partial M}{\partial p_{i}}-\frac{\partial\lambda_{j}}{\partial p_{i }}\frac{\partial M}{\partial p_{i}}]\phi_{j}+\phi_{j}^{T}\frac{\partial(K- \lambda_{j}M)}{\partial p_{i}}\frac{\partial\phi_{j}}{\partial p_{i}} \tag{35}\]It is clear that Equation (35) only requires the first-order sensitivities of the eigenvalues and eigenvectors for the calculation of the second-order eigenvalue sensitivity. The second-order sensitivity of eigenvector can be directly computed from Equation (21). Differentiating Equation (21) a second time with respect to \(p_{s}\) gives  
\[\Theta\cdot\frac{\partial^{2}\phi_{j}}{\partial p_{i}\partial p_{s}}=\frac{ \partial(\Pi\phi_{j})}{\partial p_{s}}-\frac{\partial\Theta}{\partial p_{s}} \frac{\partial\phi_{j}}{\partial p_{i}} \tag{36}\]  
Then the second-order eigenvector sensitivity \(\frac{\partial^{2}\phi_{j}}{\partial p_{i}\partial p_{s}}\) can be computed from Equation (36) as:  
\[\frac{\partial^{2}\phi_{j}}{\partial p_{i}\partial p_{s}}=\Theta^{-1}\cdot[ \frac{\partial\Pi}{\partial p_{s}}\phi_{j}+\Pi\frac{\partial\phi_{j}}{ \partial p_{s}}-\frac{\partial\Theta}{\partial p_{s}}\frac{\partial\phi_{j}}{ \partial p_{i}}] \tag{37}\]  
where  
\[\frac{\partial\Theta}{\partial p_{s}}=\frac{\partial K}{\partial p_{s}}- \frac{\partial\lambda_{j}}{\partial p_{s}}M-\lambda_{j}\frac{\partial M}{ \partial p_{s}}+\frac{\partial\lambda_{j}}{\partial p_{s}}\phi_{j}\phi_{j}^{ T}M+\lambda_{j}(\frac{\partial\phi_{j}}{\partial p_{s}}\phi_{j}^{T}M+\phi_{j} \frac{\partial\phi_{j}^{T}}{\partial p_{s}}M+\phi_{j}\phi_{j}^{T}\frac{ \partial M}{\partial p_{s}}) \tag{38}\]  
\[\frac{\partial\Pi}{\partial p_{s}}=\frac{\partial^{2}\lambda_{j}}{\partial p_ {i}\partial p_{s}}M+\frac{\partial\lambda_{j}}{\partial p_{i}}\frac{\partial M }{\partial p_{s}}+\frac{\partial\lambda_{j}}{\partial p_{s}}\frac{\partial M }{\partial p_{i}}+\lambda_{j}\frac{\partial^{2}M}{\partial p_{i}\partial p_{s }}-\frac{\partial^{2}K}{\partial p_{i}\partial p_{s}}-\frac{\partial c_{jj}} {\partial p_{s}}\lambda_{j}I-c_{jj}\frac{\partial\lambda_{j}}{\partial p_{s}}I \tag{39}\]  
From Equation (8), \(\frac{\partial c_{jj}}{\partial p_{s}}\) in Equation (39) can be computed by  
\[\frac{\partial c_{jj}}{\partial p_{s}}=\phi_{j}^{T}\frac{\partial M}{\partial p _{i}}\frac{\partial\phi_{j}}{\partial p_{s}}+\frac{1}{2}\phi_{j}^{T}\frac{ \partial^{2}M}{\partial p_{i}\partial p_{s}}\phi_{j} \tag{40}\]