# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 4 Compact Expressions for Eigenvalue and Eigenvector Jacobians
### Eigenvalue Jacobian
Specifically, let \(\mathbf{w}\) be the left eigenvector corresponding to the eigenvalue \(\lambda\) (which also has a right eigenvector \(\mathbf{v}\)),  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]2 \[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]3  
Observe, therefore, that  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]4 \[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]5 \[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]6  
Substituting this result into Equation (49) produces the following compact form that is independent of the choice of eigenvector normalization  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{\mathbf{v}^{T} }{\alpha}\otimes\left[\mathbf{v}_{0}^{H}\mathbf{A}\left(\mathbf{A}-\lambda \mathbf{I}-\frac{\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}}{\alpha}+\mathbf{N} \right)^{-1}\left(\frac{\mathbf{v}\mathbf{v}_{0}^{H}}{\alpha}-\mathbf{I} \right)+\mathbf{v}_{0}^{H}\right]. \tag{44}\]7  
This result leads to a few important observations about the sensitivity of \(\lambda\) to perturbations in \(\mathbf{A}\). First, this demonstrates that the eigenvalue Jacobian from Equation (49) is indeed the same for any choice of \(\mathbf{v}_{0}\) other than \(\mathbf{v}_{0}^{H}\mathbf{v}\approx 0\). Second, Equation (49) shows the eigenvalue Jacobian in terms of only the right eigenvector, whereas Equation (53) shows the same eigenvalue Jacobian in terms of both the right and left eigenvectors. Thus, in cases where only the right eigenvector is known (and it is not desirable to compute the corresponding left eigenvector), the result of Equation (49) provides a compact means of computing the eigenvalue Jacobian. Third, and, finally, observe the division by on Equation (53). In general, this suggests eigenvalues will be most stable when their corresponding left and right eigenvectors are colinear (as happens in a Hermitian matrix) and become unstable as the angle between \(\mathbf{v}\) and \(\mathbf{w}\) increases (\(\mathbf{w}^{H}\mathbf{v}\to 0\)).