In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of a nonsingular beta F-matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.

中文翻译：

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用异构超几何函数表示奇异 beta F 矩阵的最大特征值

在本文中，讨论了多元方差分析（MANOVA）奇异随机矩阵的最大特征值的精确分布。发展奇异随机矩阵特征值分布理论的关键是使用具有两个矩阵参数的异构超几何函数。在本研究中，我们定义了奇异 betaF-matrix 并扩展非奇异 beta 的分布F-矩阵到单数情况。我们还根据异构超几何函数给出了特征值的联合密度和最大特征值的精确分布。