We have matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ that hold observations from multivariate normal distributions. Each matrix represents $\boldsymbol{n}$ observations of $\boldsymbol{m}$ variables from a multivariable normal distribution. (The rows are the observations and each column can be considered as representing one random variable. Hence, each column in each matrix is $\boldsymbol{n}$ observations of one of the $\boldsymbol{m}$ random variables stored in that matrix.)
We can assume independence between $\boldsymbol{A}$ and $\boldsymbol{B}$ to simplify things.
$$ \boldsymbol{A_{n*m}}\sim\mathcal{N}\left(\boldsymbol{\mu_{1}},\boldsymbol{\varSigma_{1}}\right) $$ $$ \boldsymbol{B_{n*m}}\sim\mathcal{N}\left(\boldsymbol{\mu_{2}},\boldsymbol{\varSigma_{2}}\right) $$
Question One (Primary Question):
What are the distributions of $\boldsymbol{AB^{T}}$ and $\boldsymbol{A^{T}B}$?
Question Two:
What are some other useful well known properties of $\boldsymbol{AB^{T}}$ and $\boldsymbol{A^{T}B}$?
Question Three:
Are there general techniques to find properties of the product of Matrices holding Random Variable Observations from any Distribution? For example, either or both of $\boldsymbol{A}$ and $\boldsymbol{B}$ could hold observations from multivariate log normal distributions.
Please let me know if the question is not well framed or if I am not using the correct notation, or if any further details are needed.