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This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is thus an edited version with the relevant details included.

Given a random vector $X \in \mathbb{R}^k$, with a known pdf given by $f_X$. If $Y, Z \in \mathbb{R}^k$ are defined by $Y = AX$, $Z = BX$, where $A,B \in \mathbb{R}^{k\times k}$ are different, given, real-valued, singular matrices.

I know how to calculate pdfs of $Y$ and $Z$ on their own. But how do I derive the joint pdf of $Y$ and $Z$?

To be more specific:

  • $f_X$ is a mixture of $0$-mean multivariate gaussians, each component in the mixture with a different, diagonal covariance matrix (but not of the form $\Sigma = \sigma^2 I$).
  • given a unit vector $v \in \mathbb{R}^k$, my matrix $A = v v^\top$ is the projection of $X$ onto the $v$ direction, and $B = I - A$, such that $Y$ is orthogonal to $Z$.

Any help would be much appreciated.

For some context:

My goal is to check for the $f_X$, $A$ and $B$ mentioned above, whether the vectors $Y$ and $Z$ are independent. This means I need to check whether the joint distribution of $Y$ and $Z$ factorises into the product of the marginals. There are at least some cases when this is true: if, for example, $X \sim \mathcal{N}(0,\sigma^2 I)$ and $A$, $B$ were projections described above. But proving it is not true in my case would also be helpful. Hence my need to derive the joint distribution of $Y$ and $Z$.

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