On Estimating Spectral Perturbation in System Control via Extension of Bauer-Fike Theorem

The perturbation of eigenvalues constitutes a critical consideration in the domain of system control, significantly influencing the stability and performance of dynamic systems. The Bauer-Fike theorem, while powerful, is limited by its reliance on diagonalizable matrices and the condition number of eigenvector matrices, making it impractical for certain cases. To address this, we developed an algorithm using the Monte Carlo method to generate a closely related matrix more amenable to the theorem. By applying the Bauer-Fike theorem to this derived matrix, we estimate the eigenvalue variation range of the original matrix under random perturbations, thereby extending the theorem's applicability to scenarios where its direct use is constrained.