# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 3 Previous Work  
In [39], the authors of this paper presented a new technique for calculating the eigenvalue and eigenvector Jacobians. This technique allows the calculation of the full Jacobian matrices for both the eigenvalues and eigenvectors using just the eigenvalue and eigenvector being considered. Some key results from this prior work are now reviewed. Begin with the standard eigenvalue problem  
\[\mathbf{A}\mathbf{v}=\lambda\mathbf{v}, \tag{21}\]  
where \(\lambda\) is a simple eigenvalue of \(\mathbf{A}\) and \(\mathbf{v}\) is its corresponding eigenvector. Now, take the partial derivative of this relation with respect to the vectorized version of the matrix \(\mathbf{A}\) to get  
\[\left(\mathbf{A}-\lambda\mathbf{I}\right)\frac{\partial\mathbf{v}}{\partial \mathbf{A}_{\text{vec}}}=\mathbf{v}\frac{\partial\lambda}{\partial\mathbf{A}_ {\text{vec}}}-\mathbf{v}^{T}\otimes\mathbf{I}, \tag{22}\]where \(\mathbf{A}_{\text{vec}}\) is formed by stacking the columns of \(\mathbf{A}\) into a column vector and \(\otimes\) indicates the Kronecker product. This results in a single equation with two unknowns, \(\partial\mathbf{v}/\partial\mathbf{A}_{\text{vec}}\) and \(\partial\lambda/\partial\mathbf{A}_{\text{vec}}\), and thus a second equation is needed. As described in [39], the eigenvalue derivative can be calculated by considering the characteristic equation for the matrix \(\mathbf{A}\). The characteristic equation for \(\mathbf{A}\) can be expressed using the notation of exterior algebra [40] as  
\[|\lambda\mathbf{I}-\mathbf{A}|=\sum_{k=0}^{n}\lambda^{n-k}(-1)^{k}\text{Tr} \left[\wedge^{k}\mathbf{A}\right]=0, \tag{23}\]  
where \(n\) is the dimension of \(\mathbf{A}\), \(|\bullet|\) indicates the determinant, \(\wedge^{k}\mathbf{A}\) is the \(k^{th}\) exterior power of \(\mathbf{A}\), and [41]  
\[\mathbf{Q}_{k}=\left[ \begin{array}{cccc}&\text{Tr}\left[\wedge^{k}\mathbf{A }\right]=\frac{|\mathbf{Q}_{k}|}{k!}\\ \text{Tr}\left[\mathbf{A}\right]&k-1&0&\dots&0\\ \text{Tr}\left[\mathbf{A}^{2}\right]&\text{Tr}\left[\mathbf{A}\right]&k-2& \dots&0\\ \text{Tr}\left[\mathbf{A}^{3}\right]&\text{Tr}\left[\mathbf{A}^{2}\right]& \text{Tr}\left[\mathbf{A}\right]&\dots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \text{Tr}\left[\mathbf{A}^{k}\right]&\text{Tr}\left[\mathbf{A}^{k-1}\right]& \text{Tr}\left[\mathbf{A}^{k-2}\right]&\dots&\text{Tr}\left[\mathbf{A}\right] \end{array} \right]. \tag{25}\]  
With the above expression for the characteristic equation, the solution for the eigenvalue Jacobian is easy to find:  
\[\frac{\partial\lambda}{\partial\mathbf{A}_{\text{vec}}}=\frac{-\sum_{k=0}^{n} \lambda^{n-k}\frac{\partial c_{k}}{\partial\mathbf{A}_{\text{vec}}}}{\sum_{k= 0}^{n-1}c_{k}(n-k)\lambda^{n-k-1}}, \tag{26}\]  
where  
\[c_{k}=(-1)^{k}\text{Tr}\left[\wedge^{k}\mathbf{A}\right], \tag{27}\] \[\frac{\partial c_{k}}{\partial\mathbf{A}_{\text{vec}}}=\frac{(-1 )^{k}}{k}\frac{\partial\left|\mathbf{Q}_{k}\right|}{\partial\mathbf{A}_{\text {vec}}},\] (28) \[\frac{\partial\left|\mathbf{Q}_{k}\right|}{\partial\mathbf{A}_{ \text{vec}}}=\left| \begin{array}{c}\text{vec}\left(\mathbf{I}\right)^{T}\\ 2\text{vec}\left(\mathbf{A}^{T}\right)^{T}\\ 3\text{vec}\left(\left(\mathbf{A}^{2}\right)^{T}\right)^{T}\\ \vdots\\ k\text{vec}\left((\mathbf{A}^{k-1})^{T}\right)^{T}\\ \mathbf{0}_{1\times n^{2}}\\ \text{vec}\left(\mathbf{I}\right)^{T}\\ 2\text{vec}\left(\mathbf{A}^{T}\right)^{T}\\ \vdots\\ (k-1)\text{vec}\left((\mathbf{A}^{k-1})^{T}\right)^{T}\\ \mathbf{0}_{2\times n^{2}}\\ \vdots\\ \mathbf{0}_{\left(k-1\right)\times n^{2}}\\ \text{vec}\left(\mathbf{I}\right)^{T}\end{array} \right]. \tag{29}\]Now, with the independent equation for the eigenvalue Jacobian, return to Equation (22) to solve for the eigenvector Jacobian. While it would appear simple to calculate the derivative of the eigenvectors directly from Equation (22), remember that the term \(\mathbf{A}-\lambda\mathbf{I}\) is rank deficient (by the definition of an eigenvalue) and is therefore not invertible.