Random Perturbations of Matrix Polynomials

Patryk Pagacz () and Michał Wojtylak ()
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Patryk Pagacz: Jagiellonian University
Michał Wojtylak: Jagiellonian University

Journal of Theoretical Probability, 2022, vol. 35, issue 1, 52-88

Abstract: Abstract A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived, and the eigenvalues are localised. Four instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product HX of a fixed diagonal matrix H and the Wigner matrix X and two special matrix polynomials of higher degree. The results are illustrated with various examples and numerical simulations.

Keywords: Matrix polynomial; Eigenvalue; Random matrix; Limit distribution of eigenvalues; Primary 15A18; Secondary 60B20; 15B52 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-020-01048-3

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