The joint pdf of the singular values of the $m \times n$ Gaussian ensemble $X = x_{i,j} $, where the $x_{i,j} $'s are independent Normal(0,1) samples and the eigenvalues of the associated Wishart matrices $ \Sigma = XX^T$ or $X^TX $ (possibly singular) have been extensively discussed in the literature. But are there similar results for the pdf of the singular values of the random matrix $Y = AX$ where A is a known deterministic matrix, possibly diagonal.
Noted that a similar question appeared here a few years ago (also unanswered) however I then spotted in a review paper by E. Meckes, "The Eigenvalues of Random Matrices" IMAGE: December 2020 issue (number 65), that the original paper by Marchenko and Pastur$^*$ considered the eigenvalues of the ensemble $A+XTX^T$ where $T$ is diagonal and $X$ is a matrix of iid random variables. This would potentially be a solution though I don't think my math is up to disentangling their paper.
$^*$ Marchenko & Pastur, Distribution of eigenvalues for some sets of random matrices, Mathematics of the USSR-Sbornik, 1967, Volume 1, Issue 4, 457–483