Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

LSD of the Commutator of two data Matrices

Javed Hazarika, Debashis Paul

TL;DR

This work establishes the existence and precise description of the limiting spectral distributions for a random data commutator and its companion anti-commutator in high dimensional settings. Through a Bai–Silverstein–type resolvent analysis, the authors prove that the empirical eigenvalue distribution of the skew-symmetric random matrix S_n^- concentrates on the imaginary axis and is characterized by a pair of coupled Marčenko–Pastur–type equations for the Stieltjes transform s_F and the vector h = (h_1,h_2). A parallel real-axis LSD is derived for the companion S_n^+, and the approach accommodates arbitrary commuting covariance pairs, as well as important special cases (equal covariances and identity covariance) with explicit density forms. The results extend the random-matrix toolkit to skew-Hermitian settings, providing both theoretical insight and potential practical tests for covariance-structure dependence, including a density description and simulation support. The findings have implications for high-dimensional inference problems involving cross-covariance and commutator-like statistics, where asymptotic LSDs govern the behavior of spectral-based test statistics.

Abstract

We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = Σ_k^{1/2}Z_k$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $Σ_k$ are $p\times p$ positive semi-definite matrices that commute and are independent of the $Z_k$'s for $k=1,2$. We assume that $Z_k$'s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix $S_n^{-}$ will be referred to as a random commutator matrix associated with the samples $X_1$ and $X_2$. We show that, when the dimension $p$ and sample size $n$ increase simultaneously, so that $p/n \to c \in (0,\infty)$, there exists a limiting spectral distribution (LSD) for $S_n^{-}$, supported on the imaginary axis, under the assumptions that the joint spectral distribution of $Σ_1, Σ_2$ converges weakly and the entries of $Z_k$'s have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a system of Marčenko-Pastur-type functional equations. Moreover, we show that the companion matrix $S_n^{+} = n^{-1}(X_1X_2^* + X_2X_1^*)$, under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.

LSD of the Commutator of two data Matrices

TL;DR

This work establishes the existence and precise description of the limiting spectral distributions for a random data commutator and its companion anti-commutator in high dimensional settings. Through a Bai–Silverstein–type resolvent analysis, the authors prove that the empirical eigenvalue distribution of the skew-symmetric random matrix S_n^- concentrates on the imaginary axis and is characterized by a pair of coupled Marčenko–Pastur–type equations for the Stieltjes transform s_F and the vector h = (h_1,h_2). A parallel real-axis LSD is derived for the companion S_n^+, and the approach accommodates arbitrary commuting covariance pairs, as well as important special cases (equal covariances and identity covariance) with explicit density forms. The results extend the random-matrix toolkit to skew-Hermitian settings, providing both theoretical insight and potential practical tests for covariance-structure dependence, including a density description and simulation support. The findings have implications for high-dimensional inference problems involving cross-covariance and commutator-like statistics, where asymptotic LSDs govern the behavior of spectral-based test statistics.

Abstract

We study the spectral properties of a class of random matrices of the form where , 's are independent complex-valued random matrices, and are positive semi-definite matrices that commute and are independent of the 's for . We assume that 's have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix will be referred to as a random commutator matrix associated with the samples and . We show that, when the dimension and sample size increase simultaneously, so that , there exists a limiting spectral distribution (LSD) for , supported on the imaginary axis, under the assumptions that the joint spectral distribution of converges weakly and the entries of 's have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a system of Marčenko-Pastur-type functional equations. Moreover, we show that the companion matrix , under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.

Paper Structure

This paper contains 38 sections, 45 theorems, 238 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and preliminaries
  3. Stieltjes Transforms of Measures on the imaginary axis
  4. LSD under arbitrary commuting pair of scaling matrices
  5. Proof of the equivalent characterization, i.e. (\ref{['s_main_eqn_1']}) in Theorem \ref{['mainTheorem']}
  6. Sketch of the proof
  7. Assumptions
  8. Existence of Solution under Assumptions \ref{['A12']}
  9. Existence of Solution under General Conditions
  10. Properties of the L.S.D
  11. Special Case: Equal covariance matrices
  12. LSD when the common covariance is the Identity Matrix
  13. Deriving the functional roots
  14. Deriving the density of the L.S.D
  15. Simulation study
  16. ...and 23 more sections

Key Result

Lemma 3.4

For a probability distribution $F$ over the imaginary axis, let $s_{F}$ be the Stieltjes Transform (in the sense of Definition defining_StieltjesTransform). For any random variable $X \sim F$, let $\overline{F}$ represent the distribution of the real-valued random variable $-\mathbbm{i}X$. Then, the

Figures (2)

  • Figure 1: Illustration of the result of Theorem \ref{['pointMass']} as $c$ varies when $\beta = 0.7$
  • Figure 2: Simulated vs. theoretical limit distributions at various levels of $c$ for $\Sigma_n = I_p$

Theorems & Definitions (123)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 3.1
  • ...and 113 more