Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{i,j})$, In this case, the matrix $G= XX^{T}$ follows Wishart distribution and its general inverse $G^{-1}$ follows inverse non-singular Wishart  distribution with entries following zero mean and variance 
$$
G_{ij} \sim \frac{1}{\sigma_{i,j}*T(T-n-1))}
$$ from the paper [1] below.<br>
I am curious about a case where the entries of $X$, $X_{ij}\sim \mathcal{N}(\mu_{i,j},\sigma_{i,j})$. What would be the distribution of the general inverse $G^{-1}$ for such non-zero-centered data matrix?

**Reference**

[1] R. DennisCook and Liliana Forzani, "[On the mean and variance of the generalized inverse of a singular Wishart matrix](https://doi.org/10.1214/11-EJS602)" (English) 
Electronic Journal of Statistics 5, 146-158 (2011),  [MR2786485](https://mathscinet.ams.org/mathscinet-getitem?mr=2786485), [Zbl 1274.62350](https://zbmath.org/1274.62350).