# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 3 Previous Work
In order to solve this problem, we must make use of a normalization condition to make the eigenvectors unique. As discussed in the preceding section (and [34]), it is important to ensure that the normalization chosen makes the eigenvector an analytic function of \(\mathbf{A}\); therefore, choose the normalization  
\[\mathbf{v}_{0}^{H}\mathbf{v}=\alpha, \tag{30}\]  
where \(\mathbf{v}_{0}\) is any non-zero vector that is not orthogonal to \(\mathbf{v}\) and \(\alpha\) is any real non-zero scalar value. In practice, it is usually chosen that numerically \(\mathbf{v}_{0}\equiv\mathbf{v}\), as this gives rise to a more intuitive interpretation (as we discuss later); however, even when this is the case, it is still important to remember that \(\mathbf{v}_{0}\) is not a function of \(\mathbf{A}\) or \(\mathbf{v}\). In addition to making the eigenvectors analytic, Equation (30) also leads to another important relation. Consider the derivative of Equation (30) with respect to \(\mathbf{A}_{\text{vec}}\),  
\[\mathbf{v}_{0}^{H}\frac{\partial\mathbf{v}}{\partial\mathbf{A}_{\text{vec}}}= \mathbf{0}_{1\times n^{2}}, \tag{31}\]  
which shows that the normalization vector \(\mathbf{v}_{0}\) is orthogonal to the column space of the eigenvector Jacobian. Using the above normalization condition and its derivative, perform a rank-one update to the matrix on the lefthand side of Equation (22) with the so-called _null space matrix_, \(\mathbf{N}\), in order to make \(\mathbf{A}-\lambda\mathbf{I}\) invertible. This approach is discussed further in [39]. This allows for the solution of the eigenvector derivative as  
\[\frac{\partial\mathbf{v}}{\partial\mathbf{A}_{\text{vec}}}=(\mathbf{A}- \lambda\mathbf{I}+\mathbf{N})^{-1}\left[\mathbf{v}\frac{\partial\lambda}{ \partial\mathbf{A}_{\text{vec}}}-\mathbf{v}^{T}\otimes\mathbf{I}\right], \tag{32}\]  
where  
\[\mathbf{N}=\sigma\mathbf{v}_{0}\mathbf{v}_{0}^{H} \tag{33}\]  
is the Null Space Matrix and \(\sigma\) is a scaling term chosen to make \(\mathbf{N}\) approximately the same order as \(\mathbf{A}-\lambda\mathbf{I}\) (it is usually sufficient to let the scale be \(\sigma=\text{Tr}\left[\mathbf{A}\right]\)). Including the Null Space Matrix, as in Equation (33), is the same as adding a zero vector due to the relation in Equation (31). Note, of course, that a similar end objective may accomplished by using the pseudoinverse of \(\mathbf{A}-\lambda\mathbf{I}\) instead of including a Null Space Matrix (an example of this alternative approach may be found in [34]). The term \(\mathbf{A}-\lambda\mathbf{I}\) is rank deficient with a null space in the direction of \(\mathbf{v}\). Additionally, the rank one Null Space Matrix only contains information in the direction of \(\mathbf{v}_{0}\). Thus, because \(\mathbf{v}_{0}\) is required to not be orthogonal to \(\mathbf{v}\), the quantity \(\mathbf{A}-\lambda\mathbf{I}+\mathbf{N}\) is guaranteed to be full rank. For further discussion of the use of the Null Space Matrix and its properties, refer to [39]. This concludes the review of the technique proposed in [39] and attention is now turned to the simplification of these methods.