Stability of invariant subspaces of regular matrix pencils (Q1893117)

Denote by \(x\) the independent variable and let \(A\) and \(B\) be two \(n\)-by-\(n \) complex matrices. The polynomial matrix \(xB - A\) is said to be a regular (matrix) pencil if \(\text{det} (xB - A)\) is not identically zero. A complex number \(c\) is a finite eigenvalue of the regular pencil \(xB - A\) if \(\text{det} (cB - A) = 0\); infinity is an eigenvalue if \(\text{det} (B) = 0\). If \(c\) is an eigenvalue, then \(\text{Segre} (c, xB - A)\) is a nonincreasing sequence of nonnegative integers recording the elementary divisors of \(xB - A\) associated to \(c\). Let \(N\) be a subspace of the vector space of complex \(n\)-tuples written as column vectors. The authors characterize stable and Lipschitz stable invariant subspaces \(N\) for a regular pencil \(xB - A\) in terms of the sequences \(\text{Segre} (c, xB - A)\) and \(\text{Segre} (c, xB' - A')\), where \(xB' - A'\) is the restricted regular pencil associated to \(N\). The results and definitions are related to those of \textit{G. W. Stewart} and \textit{Ji-guang Sun} [Matrix perturbation theory (1990; Zbl 0706.65013)]. The proofs are based on those of \textit{I. Gohberg}, \textit{P. Lancaster}, and \textit{L. Rodman} [Invariant subspaces of matrices with applications (1986; Zbl 0608.15004)].