Eigenvalue Perturbation - Results

Results

This means it is possible to efficiently do a sensitivity analysis on as a function of changes in the entries of the matrices. (Recall that the matrices are symmetric and so changing will also change, hence the term.)

and

\frac{\partial \lambda_i}{\partial M_{(k\ell)}} = \frac{\partial}{\partial M_{(k\ell)}}\left(\lambda_{0i} + \mathbf{x}^\top_{0i} ( - \lambda_{0i}) \mathbf{x}_{0i}\right) = \lambda_i x_{0i(k)} x_{0i(\ell)}(2-\delta_k^\ell).

Similarly

and

\frac{\partial \mathbf{x}_i}{\partial M_{(k\ell)}} = -\mathbf{x}_{0i}\frac{x_{0i(k)}x_{0i(\ell)}}{2}(2-\delta_k^\ell) - \sum_{j=1\atop j\neq i}^N \frac{\lambda_{0i}x_{0j(k)} x_{0i(\ell)}}{\lambda_{0i}-\lambda_{0j}}\mathbf{x}_{0j}(2-\delta_k^\ell).

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