# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 6 Simplified Cases
### Real Diagonal Parent Matrix
#### 6.2.3 Perturbations to the Eigenspace of a Diagonalizable Matrix  
Now, reconsider the case when the parent matrix is symmetric (as was done for the previous section). Since the parent matrix is symmetric, the eigenvectors will form an orthonormal basis for \(\mathbb{R}^{n}\) and the matrix is diagonalizable as  
\[\mathbf{A}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^{T}, \tag{74}\]  
where \(\mathbf{\Lambda}\) is a diagonal matrix of the eigenvalues and \(\mathbf{V}\) is an orthogonal matrix whose columns are the eigenvectors of \(\mathbf{A}\). Substituting this into the standard eigenvalue problem gives  
\[\mathbf{V}\mathbf{\Lambda}\mathbf{V}^{T}\mathbf{v}_{i}=\lambda_{i}\mathbf{v} _{i},\]  
which can be rewritten as  
\[\mathbf{\Lambda}\mathbf{V}^{T}\mathbf{v}_{i}=\lambda_{i}\mathbf{V}^{T} \mathbf{v}_{i}. \tag{75}\]  
Since the columns of \(\mathbf{V}\) are made up of the orthogonal eigenvectors of \(\mathbf{A}\),  
\[\mathbf{V}^{T}\mathbf{v}_{i}=\mathbf{e}_{i}, \tag{76}\]  
which quickly reduces Equation (75) down to the diagonalized eigensystem,  
\[\mathbf{\Lambda}\mathbf{e}_{i}=\lambda_{i}\mathbf{e}_{i}. \tag{77}\]  
Now, suppose that the matrix \(\mathbf{A}\) is perturbed by adding a matrix \(\mathbf{\Delta}\mathbf{A}\). In the diagonalized space this matrix perturbation can be expressed as  
\[\mathbf{\Delta}\mathbf{\Lambda}=\mathbf{V}^{T}\mathbf{\Delta}\mathbf{A} \mathbf{V}, \tag{78}\]  
where \(\mathbf{\Delta}\mathbf{\Lambda}\) is an additive update to \(\mathbf{\Lambda}\). The matrix \(\mathbf{\Delta}\mathbf{\Lambda}\) will not generally be diagonal. Now that the problem has been diagonalized, the observations described above can be utilized. Since the eigenvectors in the diagonalized space form the standard basis, it is clear that the matrix \(\mathbf{\Delta}\mathbf{\Lambda}\) can be decomposed as  
\[\mathbf{\Delta}\mathbf{\Lambda}=\sum_{i=1}^{n}\sum_{k=1}^{n}\delta_{ki}\mathbf{ e}_{i}\mathbf{e}_{i}^{T}, \tag{79}\]  
where  
\[\mathbf{\Delta}\mathbf{\Lambda}=\left[ \begin{array}{cccc}\delta_{11}&\delta _{12}&\ldots&\delta_{1n}\\ \delta_{21}&\delta_{22}&\ldots&\delta_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \delta_{n1}&\delta_{n2}&\ldots&\delta_{nn}\end{array} \right] \tag{80}\]  
and  
\[\delta_{ki}=\mathbf{v}_{k}^{T}\mathbf{\Delta}\mathbf{A}\mathbf{v}_{i}. \tag{81}\]  
Now, analogous to relations in Equations (70) and (72), the perturbations in the diagonalized space are described by  
\[\mathbf{\Delta}\lambda_{i}=\delta_{ii}, \tag{82}\] \[\mathbf{\Delta}\mathbf{e}_{i}=-\sum_{k=1,k\
eq i}^{n}\frac{ \delta_{ki}}{\lambda_{k}-\lambda_{i}}\mathbf{e}_{k}. \tag{83}\]These perturbations must now be related to perturbations in the original eigenvectors, \(\mathbf{v}_{i}\). For an additive update of the diagonalized eigenvector,  
\[\mathbf{v}_{i}+\Delta\mathbf{v}_{i}=\mathbf{V}\left(\mathbf{e}_{i}+\Delta \mathbf{e}_{i}\right). \tag{84}\]  
Thus, it becomes apparent that the update to the original eigenvector is given by  
\[\Delta\mathbf{v}_{i} =\mathbf{V}\Delta\mathbf{e}_{i}\] \[=-\sum_{k=1,k\
eq i}^{n}\frac{\delta_{ki}}{\lambda_{k}-\lambda_{i }}\mathbf{v}_{k}. \tag{85}\]  
Furthermore, taking the inverse of Equation (78) combined with Equation (79) gives  
\[\Delta\mathbf{A} =\mathbf{V}\sum_{i=1}^{n}\sum_{k=1}^{n}\delta_{ki}\mathbf{e}_{i} \mathbf{e}_{i}^{T}\mathbf{V}^{T}\] \[=\sum_{i=1}^{n}\sum_{k=1}^{n}\delta_{ki}\mathbf{v}_{k}\mathbf{v}_ {i}^{T}, \tag{86}\]  
which shows that any perturbation can be expressed as a linear combination of the outer products of the eigenvectors of any symmetric matrix where the coefficients are found using Equation (81). While this approach is not efficient, it provides an interesting parallel to the modal expansion techniques discussed in [10; 11; 12; 13; 14; 15; 16; 17] as well as stability theory for eigenvectors [45]. Interestingly, the result arrived at in Equation (85) is exactly that arrived at in [45] for the symmetric case.