Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $ where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix, representing the information, and $ \bbT_{n} $ is a $ p \times p $ non-random nonnegative definite Hermitian matrix. Under some conditions on $ \bbR_n \bbR_n^* $ and $ \bbT_n $, it has been proved that for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ p $ sufficiently large. The purpose of this paper is to carry on with the study of the support of the limit spectral distribution, and we show that there is an exact separation phenomenon: with probability one, the proper number of eigenvalues lie on either side of these intervals.

翻译：令 $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $，其中 $ \bbX_n $ 为 $ p \times n $ 维矩阵，其元素为独立标准化随机变量；$ \bbR_n $ 为 $ p \times n $ 维非随机矩阵，表示信息；$ \bbT_{n} $ 为 $ p \times p $ 维非随机非负定厄米矩阵。关于 $ \bbR_n \bbR_n^* $ 与 $ \bbT_n $ 的某些条件已得到证明：在极限谱分布支撑集之外的任意闭区间内，对于所有充分大的 $ p $，几乎必然没有特征值落入该区间。本文旨在继续研究极限谱分布的支撑集，并揭示一种精确分离现象：几乎必然地，合适数量的特征值分布在这些区间的两侧。