2024/09/25 by Debashis Paul, Hazarika, Javed, Paul, Debashis · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2409.16780
openalex publication_date 2024/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the spectral properties of a class of random matrices of the form Sn- = n-1(X1 X2^* - X2 X1^*) where Xk = Σ1/2Zk, for k=1,2, Zk's are independent p× n complex-valued random matrices, and Σ is a p× p positive semi-definite matrix, independent of the Zk's. We assume that Zk's have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix Sn- will be referred to as a random commutator matrix associated with the samples X1 and X2. We show that, when the dimension p and sample size n increase simultaneously, so that p/n → c ∈ (0,∞), there exists a limiting spectral distribution (LSD) for Sn-, supported on the imaginary axis, under the assumptions that the spectral distribution of Σ converges weakly and the entries of Zk's have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a coupled Marčenko-Pastur-type functional equations. In the special case when Σ= Ip, we show that the LSD of Sn- is a mixture of a degenerate distribution at zero (with positive mass if c > 2), and a continuous distribution with a symmetric density function supported on a compact interval on the imaginary axis. Moreover, we show that the companion matrix Sn+ = Σn^(1)/(2)(Z1Z2^* + Z2Z1^*)Σn^(1)/(2), under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.