LSD of the Commutator of two data Matrices
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arXiv
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| Format: | Preprint |
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2025
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| author | Hazarika, Javed Paul, Debashis |
| author_facet | Hazarika, Javed Paul, Debashis |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00014 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | LSD of the Commutator of two data Matrices Hazarika, Javed Paul, Debashis Statistics Theory Probability 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. |
| title | LSD of the Commutator of two data Matrices |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2503.00014 |