A Perturbation Method for Index Detection for Linear Matrix Pencils
This work addresses a specific mathematical problem in linear algebra, likely incremental as it builds on existing perturbation theory for matrix pencils.
The paper tackled the problem of analyzing eigenvalue expansions and eigenvector condition numbers for linear matrix pencils under random perturbations, providing rigorous non-asymptotic bounds and error analysis, with numerical simulations to support the theoretical results.
Rigorous, non-asymptotic bounds for the Puiseux expansion of the eigenvalue at infinity are given. Error analysis is provided. Further, the expected value of the eigenvector condition number of a randomly perturbed matrix is estimated. The latter result is applied to the Cayley transform of the linear pencil. Numerical simulations illustrating the theoretical findings are provided.