de Terán, Fernando; Dopico, Froilán M.; Moro, Julio Low rank perturbation of Weierstrass structure. (English) Zbl 1163.15010 SIAM J. Matrix Anal. Appl. 30, No. 2, 538-547 (2008). Some assertions of the following types are proved: Let \(A_0 + \lambda A_1\) be a regular matrix pencil, and let \(\lambda_0\) be one of its finite eigenvalues having \(g\) elementary Jordan blocks in the Weierstrass canonical form. Then it is shown that for the most matrices \(B_0\) and \(B_1\) with \(\text{rank}(B_0+\lambda_0 B_1)< g\) there are \(g -\text{rank}(B_0+ \lambda_0 B_1)\) Jordan blocks corresponding to the eigenvalue \(\lambda_0\) in the Weierstrass form of the perturbed pencil \(A_0 + B_0 +\lambda (A_1 + B_1)\). If \(A_0 + \lambda A_1\) has an infinite eigenvalue, then the corresponding result follows from considering the zero eigenvalue of the dual pencils \(A_1 + \lambda A_0\) and \(A_1 + B_1 + \lambda (A_0 + B_0)\). Reviewer: Grozio Stanilov (Sofia) Cited in 24 Documents MSC: 15A22 Matrix pencils 15A18 Eigenvalues, singular values, and eigenvectors 15A21 Canonical forms, reductions, classification Keywords:regular matrix pencils; Weierstrass canonical form; low rank perturbations; matrix spectral perturbation theory; eigenvalues; Jordan blocks PDFBibTeX XMLCite \textit{F. de Terán} et al., SIAM J. Matrix Anal. Appl. 30, No. 2, 538--547 (2008; Zbl 1163.15010) Full Text: DOI