Spectral properties of random triangular matrices

TL;DR

This paper investigates the spectral distribution of symmetric and non-symmetric triangular patterned matrices, establishing their limiting spectral distribution and moments, with specific focus on the symmetric triangular Wigner matrix.

Contribution

It proves the existence of the LSD for symmetric triangular matrices and derives moments for the symmetric triangular Wigner matrix using Catalan words, advancing understanding of spectral properties.

Findings

LSD exists for symmetric triangular matrices

Explicit moments derived for symmetric triangular Wigner matrix

Challenges remain in deriving explicit formulas for other patterned matrices

Abstract

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.