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Abstract
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), 9711004, 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 socalled eigenvalue exclusion theorem are strong linearizations for P(z). This property and the blocksymmetric 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 on the representation of the pencils in DL(P) via Bézoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181209, 2015].
Original language  English 

Pages (fromto)  88131 
Number of pages  44 
Journal  Linear Algebra and Its Applications 
Volume  677 
DOIs  
Publication status  Published  15 Nov 2023 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Bézout matrix
 Bézoutian
 DL(P)
 Eigenvalue exclusion theorem
 Linearization
 Minimal basis
 Minimal indices
 Root polynomial
 Singular matrix polynomial
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 1 Finished

Noferini_Vanni_AoF_Project: Noferini Vanni Academy Project
Noferini, V., Quintana Ponce, M., Barbarino, G., Wood, R. & Nyman, L.
01/09/2020 → 31/08/2024
Project: Academy of Finland: Other research funding