Limiting spectral distribution of the commutator of two data matrices

In this paper, we study the spectral properties of a class of random matrices of the form [Formula: see text], where [Formula: see text], [Formula: see text]’s are independent [Formula: see text] complex-valued random matrices, and [Formula: see text] are [Formula: see text] positive semi-definite matrices that commute and are independent of the [Formula: see text]’s for [Formula: see text]. We assume that [Formula: see text]’s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix [Formula: see text] will be referred to as a random commutator matrix associated with the samples [Formula: see text] and [Formula: see text]. We show that, when the dimension [Formula: see text] and sample size [Formula: see text] increase simultaneously, so that [Formula: see text], there exists a limiting spectral distribution (LSD) for [Formula: see text], supported on the imaginary axis, under the assumptions that the joint spectral distribution of [Formula: see text] converges weakly. 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 [Formula: see text], under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.