Mackey, D. Steven; Mackey, Niloufer; Mehl, Christian; Mehrmann, Volker Vector spaces of linearizations for matrix polynomials. (English) Zbl 1132.65027 SIAM J. Matrix Anal. Appl. 28, No. 4, 971-1004 (2006). Companion form linearizations do not reflect any structure that may be present in the original polynomial \(P\), so that their use in numerics may be problematic. Another issue is conditioning, and numerics may produce different results for each linearization. A further issue for linearizations concerns eigenvalues at infinity, as they occur in circuit simulation and wave guides. The authors show how to systematically generate, for any regular matrix polynomial, large classes of linearizations that address those issues. This is done by using a natural generalization of companion forms, resulting in two vector spaces of pencils, with an elegant characterization of exactly which pencils in the intersection of the two spaces are linearizations for \(P\). The eigenvectors of any such pencil are rather simply related to those of \(P\). Reviewer: E. Kreyszig (Ottawa) Cited in 10 ReviewsCited in 145 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A22 Matrix pencils 15A54 Matrices over function rings in one or more variables Keywords:matrix polynomial; matrix pencil; linearization; strong linearization; shifted sum; companion form; eigenvalues; eigenvectors PDFBibTeX XMLCite \textit{D. S. Mackey} et al., SIAM J. Matrix Anal. Appl. 28, No. 4, 971--1004 (2006; Zbl 1132.65027) Full Text: DOI Link