# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 5 On the Choice of a Normalization Vector
In light of the above discussion, we stress again that \(\mathbf{v}_{0}\) is not the same as \(\mathbf{v}\). These results simply demonstrate that the best numerical performance is achieved when the values of the arbitrary vector \(\mathbf{v}_{0}\) are set to be the same as \(\mathbf{v}\). Consequently, one may wonder how the usual normalization choice of \(\mathbf{v}^{H}\mathbf{v}=\alpha\) (instead of the \(\mathbf{v}_{0}^{H}\mathbf{v}=\alpha\) normalization suggested in this paper) affects the Jacobians. Observation of the problem geometry reveals that the normalization proposed in Equation (30) (\(\mathbf{v}_{0}^{H}\mathbf{v}=\alpha\)) serves as a good approximation for the usual normalization of \(\mathbf{v}^{H}\mathbf{v}=\alpha\) when \(\mathbf{A}\) is well conditioned and the values of \(\mathbf{v}_{0}\) are set to that of \(\mathbf{v}\). This is because the usual normalization constrains the eigenvectors to fall on the hypersphere of radius \(\sqrt{\alpha}\), while the normalization from Equation (30)  
Figure 2: Histograms of the condition number of the term \(\mathbf{A}-\lambda\mathbf{I}-\mathbf{v}\mathbf{v}_{0}^{H}\mathbf{A}/\alpha\) normalized by the number of dimensions, \(n\), compared to [PERSON]’s probability density function for the condition number of \(\mathbf{A}\). Here, \(\mathbf{A}\) is a real matrix. constrains the perturbed eigenvectors to fall on the hyperplane tangent to the hypersphere of radius \(\alpha/\left\|\mathbf{v}\right\|\) at \(\mathbf{v}_{0}\). Therefore, as long as the perturbation of \(\mathbf{A}\) leaves the perturbed eigenvector near the original eigenvector, the proposed normalization approximates the normalization constraining the eigenvectors to the hypersphere to first order. Furthermore, when \(\mathbf{v}_{0}\) is chosen to be \(\mathbf{v}\), the difference between the usual normalization and Equation (30) rarely needs to be considered in practice except in very rare situations, such as when attempting to numerically verify the analytic expressions as was done in the \"Numerical Validation\" section of this paper. With these observations in mind, additional remarks are necessary regarding the expressions in the literature that rely on the normalization \(\mathbf{v}^{H}\mathbf{v}=\alpha\). Recall from our earlier discussion that using \(\mathbf{v}^{H}\mathbf{v}=\alpha\) does not result in a valid expression for the eigenvector derivatives because the normalization is not analytic. Despite this, the numerical results produced by these methods will be equivalent to the numerical results of the expressions developed in this paper when \(\mathbf{v}_{0}\) is chosen to be \(\mathbf{v}\). While the expressions for the derivatives are numerically equivalent, the derivatives from the existing literature are not valid for the normalization employed. This means that that the derivatives in the existing literature cannot be used to predict the perturbations caused to eigenvectors by perturbations to \(\mathbf{A}\) when the eigenvectors are complex. The derivatives and normalization in this paper are valid for all complex eigenvectors and can be used to predict the perturbation caused to the eigenvectors by the perturbations to \(\mathbf{A}\).