# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 4 Compact Expressions for Eigenvalue and Eigenvector Jacobians
### Eigenvalue Jacobian  
Now, consider how the eigenvector Jacobian may be used to find the eigenvalue Jacobian. Substitute the result of Equation (41) into Equation (36) and collect like terms  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}\right)^{-1} \left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I}\right)+\mathbf{v} _{0}^{H}\right]. \tag{43}\]  
For the zero-eigenvalue case, the equivalent expression for the eigenvalue Jacobian becomes  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]  
Focusing on the general case from Equation (43), we observe that further simplification is possible. Begin simplification by recognizing that the following are true:  
\[\left(\mathbf{A}-\lambda\mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0 }^{H}\mathbf{A}}{\alpha}\right)\mathbf{v} =\left(\mathbf{A}-\lambda\mathbf{I}\right)\mathbf{v}-\left(\frac {\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}\right)\mathbf{v}\] \[=-\frac{1}{\alpha}\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}\mathbf{v}\] \[=-\frac{\lambda}{\alpha}\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{v}\] \[=-\lambda\mathbf{v} \tag{45}\]  
and, therefore,  
\[\left(\mathbf{A}-\lambda\mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{ A}}{\alpha}\right)^{-1}\mathbf{v}=-\frac{1}{\lambda}\mathbf{v}. \tag{46}\]Applying this result, the intermediate term in Equation (43) may be simplified  
\[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda\mathbf{I}- \frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}\right)^{-1}\left(\frac{ \mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I}\right)\] \[=-\frac{\mathbf{v}_{0}^{H}\mathbf{Av}\mathbf{v}_{0}^{H}}{\lambda \alpha}-\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda\mathbf{I}-\frac{ \mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}\right)^{-1} \tag{47}\] \[=-\frac{\mathbf{v}_{0}^{H}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{v} _{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda\mathbf{I}-\frac{\mathbf{v}\mathbf{ v}_{0}^{H}\mathbf{A}}{\alpha}\right)^{-1}\] \[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]0  
which, upon substitution into Equation (43), leads to  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]1  
providing a concise expression for \(\partial\lambda/\partial\mathbf{A}_{\text{vec}}\) only in terms of the right eigenvector and the chosen normalization convention, \(\mathbf{v}_{0}^{H}\mathbf{v}=\alpha\). Intuitively, however, we expect the eigenvalue Jacobian to be unrelated to the choice of eigenvector normalization. Indeed, we find \(\partial\lambda/\partial\mathbf{A}_{\text{vec}}\) to be the same for any choice of \(\mathbf{v}_{0}\), so long as \(\mathbf{v}_{0}^{H}\mathbf{v}\
eq 0\). We may show this by rewriting Equation (49) without the use of \(\mathbf{v}_{0}\) or \(\alpha\), although doing so does require consideration of the left eigenvector.