I have been trying to derive some conditional distribution for parameters of a linear transformation (represented as a matrix) and I had a lot of help on this thread yesterday. However, I realised I did something which could be a terrible mistake.

So, I have vector valued observation $y$, which is modelled as a linear combination of another observation $x$.

The way $y$ is modelled is:

$$ y \sim \mathrm{N} (Ax, \Sigma). $$

The thing to note here is that $A$ is a matrix. Now, I wanted to put a normal prior on the transformation parameters i.e. the entries of $A$ and I do that as:

$$ A \sim \mathrm{N} (A_0, \nabla). $$

Now, here I see $A$ as a vector. Now, when I tried to get the conditional distribution of $A$ by multiplying the two Gaussians, I ran into a bit of trouble because of this matrix-vector discrepancy. I could not separate the terms properly as was suggested by Glen_b in that thread.

I was wondering if there is a way to deal with this so that I can still derive the conditional distribution in a closed form way. Perhaps, what I have done is valid and I need to find some linear algebra tricks to make this work. The reply on the last thread was very useful but then I realised that this was perhaps a mistake on my part to have this model. Although I am not sure if I am really wrong.