The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

Giovanni Barbarino*, Vanni Noferini

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex n× n Gaussian monic matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n→ ∞ with k constant and (2) k→ ∞ with n constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: We slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a nonzero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.

Original languageEnglish
Publication statusE-pub ahead of print - 16 Feb 2022
MoE publication typeA1 Journal article-refereed


  • Companion matrix
  • Empirical spectral distribution
  • Polynomial eigenvalue problem
  • Random matrix polynomial
  • Strong circle law


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