  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.
