In this paper, we investigate the spectral properties of the sample canonical correlation (SCC) matrix under the alternative hypothesis to provide a more comprehensive description of the association between two sets of variables. Our research involves establishing the relationship between the eigenvalues of the SCC matrix and the block correlation matrix, as well as proving the universality of the Stieltjes transform of the limiting spectral distribution (LSD) of the block correlation matrix. By combining the results from the normal case, we establish the limiting spectral distribution (LSD) of the SCC matrix with a general underlying distribution under the arbitrary rank alternative hypothesis. Finally, we present several simulated examples and find that they fit well with our theoretical results.

翻译：本文研究了备择假设下样本典型相关（SCC）矩阵的谱性质，旨在更全面地描述两组变量之间的关联性。我们的工作包括建立SCC矩阵特征值与分块相关矩阵之间的关系，并证明分块相关矩阵极限谱分布（LSD）的Stieltjes变换的普适性。结合正态情形下的结果，我们建立了任意秩备择假设下一般分布背景的SCC矩阵的极限谱分布（LSD）。最后，我们给出若干模拟实例，发现其与理论结果吻合良好。