Linear Transformation of Dirichlet Distribution

Question

I am wondering what is the best way to represent the probability distribution of a linearly transformed Dirichlet distribution:

let {X = [x₁, x₂ ... x_n] | Σx_i = 1} be a set of multinomial probabilities that follow a Dirichlet distribution D_ɑn.

Consider matrix A_nxm where:
A_nxmX = Y st. (Σy_i = 1 ∀X) (Note A_nxm is derived from a Bayesian net)

If X follows a Derichlet Distribution what distribution does Y follow?!?

If believe that if A_nxm is an invertible square matrix. By A_nxmX being injective and surjective, Y follows the below distribution:
Likelihood(Y) = D_ɑn(A^-1Y)

But I am not sure what to do for other cases.
Thank you!

this paper may be useful but it is beyond me...
https://onlinelibrary.wiley.com/doi/epdf/10.2307/3315988