General distribution generator, Poisson distribution and correlation?

Context: I am a PhD student in physics, and I am trying to simulate some physical processes with particular statistics.

In particular, I have two matrices. The first matrix, A, has a Poisson distribution, i.e. var(A)=eta*u, where eta and u are two parameters of the experiment, and var is the variance of the matrix.

Now I want to retrieve a second matrix, B, that has a Poisson distribution var(B)=u, but at the same time, it is correlated with the first one, such as corr(A,B)=eta*u, where corr is the correlation (for example the Pearson correlation coefficient)

Can I retrieve the matrix B in some way? I know the function poissrnd, but here I also have a restraint considering the correlation. Is there any function that can do this?

Answer 1

[the cyclist]

One way to generate correlated variables from non-normal distributions is to use copulas. There is a detailed discussion on this documentation page. I haven't thought carefully about your specific use case, but I expect it will work.