On the invariance of the sign pattern of matrix eigenvectors under perturbation (Q1316186)

The authors investigate two problems concerning a real regular matrix \(A\) with simple eigenvalue \(\lambda_ i\) and corresponding eigenvector \(x^ i\). Moreover, let \(\delta A\) be a real perturbation of \(A\). Firstly, if \(\delta A\) is satisfying certain explicitly given bounds then the eigenvector \(\widetilde x^ i\) of \(A+\delta A\) will have the same sign pattern as \(x^ i\). Secondly, given a real interval \(I\) with midpoint \(\lambda_ i\) then conditions on \(\delta A\) are stated ensuring that the eigenvalue \(\widetilde \lambda_ i\) of \(A + \delta A\) is in \(I\) and that the associated eigenvector \(\widetilde x^ i\) has the same sign pattern as \(x^ i\).