The DL(P) vector space of pencils for singular matrix polynomials

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Linear Algebra and Its Applications, Volume 677
Given a possibly singular matrix polynomial P(z), we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space DL(P) introduced in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl. 28(4), 971-1004, 2006] are related to those of P(z). If P(z) is regular, it is known that those pencils in DL(P) satisfying the generic assumptions in the so-called eigenvalue exclusion theorem are strong linearizations for P(z). This property and the block-symmetric structure of the pencils in DL(P) have made these linearizations among the most influential for the theoretical and numerical treatment of structured regular matrix polynomials. However, it is also known that, if P(z) is singular, then none of the pencils in DL(P) is a linearization for P(z). In this paper, we prove that despite this fact a generalization of the eigenvalue exclusion theorem holds for any singular matrix polynomial P(z) and that such a generalization allows us to recoverall the relevant quantities of P(z) from any pencil in DL(P) satisfying the eigenvalue exclusion hypothesis. Our proof of this general theorem relies heavily on the representation of the pencils in DL(P) via Bézoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181-209, 2015].
Funding Information: FD is supported by grant PID2019-106362GB-I00 funded by MCIN/AEI/10.13039/ 501100011033 and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors ( EPUC3M23 ), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation). VN is supported by an Academy of Finland grant ( Suomen Akatemian päätös 331240 ). Publisher Copyright: © 2023 The Author(s)
Bézout matrix, Bézoutian, DL(P), Eigenvalue exclusion theorem, Linearization, Minimal basis, Minimal indices, Root polynomial, Singular matrix polynomial
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Dopico , F M & Noferini , V 2023 , ' The DL(P) vector space of pencils for singular matrix polynomials ' , Linear Algebra and Its Applications , vol. 677 , pp. 88-131 .