Sensitivity and Computation of a Defective Eigenvalue
Zhonggang Zeng · SIAM Journal on Matrix Analysis and Applications · 2016
A defective eigenvalue is well documented to be hypersensitive to data perturbations and round-off errors, making it a formidable challenge in numerical computation particularly when the matrix is known through approximate data. This paper establishes a finitely bounded sensitivity of a defective eigenvalue with respect to perturbations that preserve the geometric multiplicity and the smallest Jordan block size. Based on this perturbation theory, numerical computation of a defective eigenvalue is regularized as a well-posed least squares problem so that it can be accurately carried out using floating point arithmetic even if the matrix is perturbed.