Generic rank-two perturbations of structured regular matrix pencils (Q2807239)

Let \(E\) and \(A\) be two \(n\times n\) complex matrices where \(E\) is symmetric and \(A\) is skew symmetric; then the pencil \(P_{\lambda}=\lambda E-A\) is called an even \(T\)-alternating pencil. If \(\Delta E\) and \(\Delta A\) (symmetric and skew-symmetric, respectively) are perturbations of \(E\) and \(A\), then the perturbation \(\Delta P_{\lambda}=\lambda\Delta E-\Delta A\) is said to have normal rank \(k\) if \(\Delta P_{\lambda}\) has rank \(k\) for all but finitely many \(\lambda\). The author is interested in studying how the sizes of the Jordan blocks are changed when \(P_{\lambda}\) is replaced by \(P_{\lambda}+\Delta P_{\lambda}\) and \(\Delta P_{\lambda}\) has small normal rank. He considered the case of normal rank \(1\) (cf. [\textit{L. Batzke}, Linear Algebra Appl. 458, 638--670 (2014; Zbl 1294.15011)]), and in this paper, he investigates the case of normal rank \(2\). He summarizes his results as follows, when \(\Delta P_{\lambda}\) is of normal rank \(2\). At each eigenvalue \(\hat{\lambda}\,\)of \(P_{\lambda}\), the Jordan structure of \(P_{\lambda}+\Delta P_{\lambda}\) is generically that of \(P_{\lambda}\) except for the following changes: (1) the largest two of the Jordan blocks corresponding to \(\hat{\lambda}\) are destroyed; (2) if the largest Jordan block at \(\hat{\lambda}\) is unpaired and the second largest block is paired to an identical one, this largest remaining block will grow by one in size; (3) if \(\hat{\lambda}\) is a single (or double) eigenvalue of \(\Delta P_{\lambda}\), then one (respectively, two) new Jordan blocks of size one are created at \(\hat{\lambda}\). Similar results are obtained for odd \(T\)-alternating pencils (where \(E\) is skew-symmetric and \(A\) symmetric) and palindromic matrix pencils (pencils of the form \(\lambda B\pm B^{T}\)).