On the Spectral Norms of Pseudo-Wigner and Related Matrices

2017/08/14 by Soloveychik, Ilya, Tarokh, Vahid
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1708.04291

Abstract

We investigate the spectral norms of symmetric N × N matrices from two pseudo-random ensembles. The first is the pseudo-Wigner ensemble introduced in "Pseudo-Wigner Matrices" by Soloveychik, Xiang and Tarokh and the second is its sample covariance-type analog defined in this work. Both ensembles are defined through the concept of r-independence by controlling the amount of randomness in the underlying matrices, and can be constructed from dual BCH codes. We show that when the measure of randomness r grows as N^ρ, where ρ∈ (0,1] and ε > 0, the norm of the matrices is almost surely within o(\fraclog^1+ε NN^min[ρ,2/3]) distance from 1. Numerical simulations verifying the obtained results are provided.

Related

• Pseudo-Wigner Matrices
• Bounds on the norm of Wigner-type random matrices
• Random matrices: The Universality phenomenon for Wigner ensembles
• Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices
• Random Matrices from Linear Codes and Wigner's semicircle law