Perturbation of null spaces with application to the eigenvalue problem and generalized inverses (Q1399918)

This paper concerns \[ A(\varepsilon) x(\varepsilon)= \lambda(\varepsilon) x(\varepsilon)\tag{1} \] and in particular \[ A(\varepsilon)= \sum^\infty_{k=0} \varepsilon^k A_k,\quad A_k\in\mathbb{R}^{n\times n},\tag{2} \] where (2) converges for \(|\varepsilon|\leq R\), \(R> 0\), and gives Taylor series for the eigenvectors that constitute a basis for the perturbed null space. This is applied to the calculation of Laurent series for the perturbed group inverse and pseudoinverse matrices. A few formal and one numerical examples are included.