# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 2 Existence of Eigenvalue and Eigenvector Jacobians  
The standard algebraic eigenvalue problem is defined as  
\[\mathbf{A}\mathbf{v}=\lambda\mathbf{v}, \tag{1}\]  
where \(\mathbf{A}\in\mathbb{C}^{n\times n}\) is a \(n\times n\) square matrix, \(\mathbf{v}\in\mathbb{C}^{n}\) is an eigenvector of the system, and \(\lambda\in\mathbb{C}\) is an eigenvalue of the system. As should be apparent, each eigenvalue and eigenvector pair is only implicitly defined according to  
\[\left(\mathbf{A}-\lambda\mathbf{I}\right)\mathbf{v}=\mathbf{0}. \tag{2}\]  
It is well known that eigenvectors corresponding to simple eigenvalues are only unique when a constraint is placed on their norm (i.e., \"length\"). In addition, the eigenvector normalization can be expressed implicitly as  
\[\boldsymbol{x}\mathbf{v}-\alpha=0, \tag{3}\]  
where \(\boldsymbol{x}\) is a row vector formed by choosing a column normalization vector and transposition method and \(\alpha\) is a normalization constant. Although the choice of \(\alpha\) is arbitrary, we often choose \(\alpha=1\). These two implicit vector functions can be combined to form a single implicit vector equation:  
\[\mathbf{f}(\mathbf{A},\mathbf{y})=\left[\begin{array}{c}\left(\mathbf{A}- \lambda\mathbf{I}\right)\mathbf{v}\\ \boldsymbol{x}\mathbf{v}-\alpha\end{array}\right]=\mathbf{0}_{n+1\times 1}, \tag{4}\]  
where \(\mathbf{y}=\left[\begin{array}{cc}\mathbf{v}^{T}&\lambda\end{array}\right]^ {T}\). To determine if eigenvalue and eigenvector derivatives exist with respect to the elements of \(\mathbf{A}\), we must determine if both \(\lambda\) and \(\mathbf{v}\) can be expressed as an explicit function of \(\mathbf{A}\). The implicit function theorem can be used to test if this is possible.  
The implicit function theorem states that as long as the determinant of the Jacobian of \(\mathbf{f}\) with respect to \(\mathbf{y}\) can be found and is nonzero, then \(\mathbf{y}\) can be expressed as an explicit function of \(\mathbf{x}\). Therefore, consider the Jacobian of \(\mathbf{f}\) as defined in Equation (4):  
\[\mathbf{J}=\frac{\partial\mathbf{f}}{\partial\mathbf{y}}=\left[\begin{array}[] {cc}\mathbf{A}-\lambda\mathbf{I}&-\mathbf{v}\\ \frac{\partial}{\partial\mathbf{v}}(\boldsymbol{x}\mathbf{v})&0\end{array} \right], \tag{5}\]  
where \(\frac{\partial}{\partial\mathbf{v}}(\boldsymbol{x}\mathbf{v})\) will be a \(n\times 1\) row vector from the rules of matrix vector calculus. Recognizing the block structure of the Jacobian, it can be shown that the determinant is given by  
\[\left|\mathbf{J}\right| =-1\left|\mathbf{A}-\lambda\mathbf{I}\right|+\left|\mathbf{A}- \lambda\mathbf{I}+\mathbf{v}\left[\frac{\partial}{\partial\mathbf{v}}( \boldsymbol{x}\mathbf{v})\right]\right|\] \[=\left|\mathbf{A}-\lambda\mathbf{I}+\mathbf{v}\left[\frac{ \partial}{\partial\mathbf{v}}(\boldsymbol{x}\mathbf{v})\right]\right|. \tag{6}\]  
Therefore, in order to evaluate the determinant of the Jacobian, we must evaluate \(\partial(\boldsymbol{x}\mathbf{v})/\partial\mathbf{v}\) for the specific normalization described by the row vector \(\boldsymbol{x}\). The following subsections discuss two potential eigenvector normalizations.