Let $\mathbf{A}$ be an $m\times n$ random matrix with entries $A_{ij}$ being jointly Gaussian. Suppose all of these variables are independent of the random vector $\mathbf{X} = (X_1,\ldots,X_n)^\top$ which follows a multivariate Gaussian distribution.

Does the random vector $\mathbf{Y} = \mathbf{A}\mathbf{X}$ follow any particular distribution? Maybe some sort of modified $\chi^2$-distribution? I'm mostly interested in some expression for the covariance matrix of $\mathbf{Y}$ that can hopefully be expressed in terms of the the expected values and covariances associated with $\mathbf{A}$ and $\mathbf{X}$. For simplicity, it is okay to assume $\mathbf{X}$ to have mean zero.

Any guidance is appreciated.

For instance, I realized that, for $m=n=1$, we have \begin{align*} \text{var}(AX) &= \mathbb{E}(A^2)\mathbb{E}(X^2) - \mathbb{E}^2(A)\mathbb{E}^2(X)\\ &= (\text{var}(A)+\mathbb{E}^2(A))(\text{var}(X)+\mathbb{E}^2(X)) - \mathbb{E}^2(A)\mathbb{E}^2(X). \end{align*}