A Kronecker Matrix Inequality with a Statistical Application

Solutions. An elegant and short solution to the original problem was proposed by Heinz Neudecker and Albert Satorra, the posers of the problem. Independently, Götz Trenkler proposed a solution that shows that the problem could be modified into an "if and only if" statement. Finally, Heinze Neudecker and Shuangzhe Liu, who saw Trenkler's solution, also proposed an elegant and short solution about the necessity of the condition. All three solutions are published below.