LSD of the Commutator of two data Matrices

We study the spectral properties of a class of random matrices of the form $S_n^{-}=n^{-1}(X_1{X}_2^{\ast }-X_2{X}_1^{\ast })$ where $X_k={\mathrm{\Sigma }}_k^{1\mathrm{/}2}{Z}_k$, $Z_k$’s are independent $p\times n$ complex-valued random matrices, and ${\mathrm{\Sigma }}_k$ 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 $p\mathrm{/}n\to c\in (0,\mathrm{\infty })$, there exists a limiting spectral distribution (LSD) for $S_n^{-}$, supported on the imaginary axis, under the assumptions that the joint spectral distribution of ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ 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 $S_n^{+}=n^{-1}(X_1{X}_2^{\ast }+X_2{X}_1^{\ast })$, under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.