On the Spectral Norms of Pseudo-Wigner and Related Matrices

TL;DR

This paper analyzes the spectral norms of pseudo-random symmetric matrices from two ensembles, showing they are close to 1 with high probability as matrix size grows, depending on the randomness measure.

Contribution

It introduces bounds on the spectral norms of pseudo-Wigner and covariance-type matrices constructed from dual BCH codes, extending understanding of their spectral behavior.

Findings

Spectral norms are within a small distance from 1 as N grows.

The bounds depend on the growth rate of the randomness parameter r.

Numerical simulations confirm theoretical results.

Abstract

We investigate the spectral norms of symmetric $N \times N$ 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 $N^\rho$, where $\rho \in (0,1]$ and $\varepsilon > 0$, the norm of the matrices is almost surely within $o\left(\frac{\log^{1+\varepsilon} N}{N^{\min[\rho,2/3]}}\right)$ distance from $1$. Numerical simulations verifying the obtained results are provided.