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A Mistake on the Computation of Jacobians of Singular Random Matrices

Shiqing Wang, Limin Su


In the calculation of Jacobians of singular random matrices, Uhlig (On singular Wishart and singular multivariate beta distributions. Ann. Stat. 22 395-405. 1994) and Diaz-Garcia (Proof of conjectures of H. Uhlig on the singular multivariate Beta and the Jacobian of a certain matrix transformation. Ann. Stat. 25 2018-2023. 1997) gave a series of theorems. These theorems became a foundation on the research of the distributions of singular random matrices. In this paper, firstly, some counter-examples are given; they show that the some of Uhlig’s and Diaz-Garcia’s theorems are incorrect. This means that the literatures (Uhlig 1994; Diaz-Garcia et al. 1997a, 1997b, 2005a, 2005b, 2006, 2007, 2008) are problematical because they used these theorems. Secondly, the reasons which cause these theorems wrong are found, and the revision and the extension of a series of classical theorems are given for the calculation of Jacobians of nonsingular random matrices. Finally, the calculation of Jacobians of singular random matrices is discussed, and some new results are obtained.


Jacobian of transformation, singular random matrix, exterior product, exterior differential, functionally independent.

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