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

12/16/2022

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by   Froilán Dopico, et al.

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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 recover all 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 in 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].