# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 2 Existence of Eigenvalue and Eigenvector Jacobians
### Constraining the Eigenvectors to a Hyperplane  
Now, consider an eigenvector normalization which constrains the length of the projection of \(\mathbf{v}\) onto an arbitrary vector, \(\mathbf{v}_{0}\) (so long as \(\mathbf{v}_{0}^{H}\mathbf{v}\
eq 0\), where we stress here that \(\mathbf{v}_{0}\) is not functionally dependent on \(\mathbf{v}\), \(\lambda\), or \(\mathbf{A}\).) Such a normalization occurs when \(\mathbf{x}=\mathbf{v}_{0}^{H}\) and constrains \(\mathbf{v}\) to lie on a hyperplane. Therefore, substituting this into Equation (3) leads to:  
\[\mathbf{v}_{0}^{H}\mathbf{v}=\alpha\qquad\rightarrow\qquad-\alpha+\mathbf{v}_ {0}^{H}\mathbf{v}=0. \tag{13}\]  
Specifically, note that the eigenvector \(\mathbf{v}\) is now constrained to lie on a hyperplane tangent to the hypersphere of radius \(\alpha/\left\|\mathbf{v}\right\|\) at the point \(\mathbf{v}_{0}\). Expressing this normalization in the terms of the elements of \(\mathbf{v}\) and \(\mathbf{v}_{0}\) gives  
\[-\alpha+\sum_{k=1}^{n}\overline{v_{0k}}v_{k} =-\alpha+\sum_{k=1}^{n}(a_{k}-ib_{k})(x_{k}+iy_{k})\] \[=-\alpha+\sum_{k=1}^{n}a_{k}x_{k}+b_{k}y_{k}+i(a_{k}y_{k}-b_{k}x_ {k}), \tag{14}\]where \(v_{k}\) is as defined before and \(v_{0k}=a_{k}+ib_{k}\) is the \(k^{th}\) element of \(\mathbf{v}_{0}\).  
As before, since we are seeking a complex partial derivative, the Cauchy-Riemann equations are used to check if the partial derivative of the normalization is analytic in a region around the eigenvector. The partial derivatives with respect to the components of the \(j^{th}\) elements of the eigenvector are  
\[\frac{\partial}{\partial x_{j}}\text{Re}\left[\sum_{k=1}^{n}a_{k}x_{k}+b_{k}y _{k}+i(a_{k}y_{k}-b_{k}x_{k})\right]\!\!=a_{j}, \tag{15}\]  
\[\frac{\partial}{\partial x_{j}}\text{Im}\left[\sum_{k=1}^{n}a_{k}x_{k}+b_{k}y _{k}+i(a_{k}y_{k}-b_{k}x_{k})\right]\!\!=-b_{j}, \tag{16}\]  
and  
\[\frac{\partial}{\partial y_{j}}\text{Re}\left[\sum_{k=1}^{n}a_{k}x_{k}+b_{k}y _{k}+i(a_{k}y_{k}-b_{k}x_{k})\right]\!\!=b_{j}, \tag{17}\]  
\[\frac{\partial}{\partial y_{j}}\text{Im}\left[\sum_{k=1}^{n}a_{k}x_{k}+b_{k}y _{k}+i(a_{k}y_{k}-b_{k}x_{k})\right]\!\!=a_{j}, \tag{18}\]  
which satisfy the Cauchy-Riemann equations for any choice of \(a_{j}\) and \(b_{j}\) indicating that the normalization is analytic in all of \(\mathbb{C}^{n}\). Since the Cauchy-Riemann equations are satisfied, the vector form of the partial derivative of the eigenvector normalization constraint with respect to the eigenvector is given as  
\[\frac{\partial}{\partial\mathbf{v}}\left(\mathbf{v}_{0}^{H}\mathbf{v}-\alpha \right)=\mathbf{v}_{0}^{H} \tag{19}\]  
using the rules of matrix vector calculus.  
This allows the implicit function theorem to be used to check to see if the eigenvalues and eigenvectors are guaranteed to be analytic functions of \(\mathbf{A}\) on some domain centered at their nominal values. Substituting Equation (19) into Equation (6) gives  
\[|\mathbf{J}|=\left|\mathbf{A}-\lambda\mathbf{I}+\mathbf{v}\mathbf{v}_{0}^{H} \right|, \tag{20}\]  
which is guaranteed to be full rank and have a non-zero determinant if \(\lambda\) is a simple eigenvalue (defined as an eigenvalue with a multiplicity of 1). Therefore, using the implicit function theorem with this normalization, it is guaranteed that the eigenvectors and eigenvalues are analytic functions of \(\mathbf{A}\) in some domain around their nominal values for simple eigenvalues. Due to this, the rest of this paper will use the normalization \(\mathbf{v}_{0}^{H}\mathbf{v}=\alpha\) and the case when the eigenvalue is simple.