Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for testing the equality of a covariance matrix, equality of a mean vector and a covariance matrix, independence ...

The first is a Bernoulli random null model in which the null matrix has the same total number of interactions as the original matrix, albeit randomly positioned.

The Null Ideal of a Matrix: 11083 William P. Wardlaw and Jean-Pierre Grivaux The American Mathematical Monthly , p. 371 (1 page) ...