Given a correlation matrix $C_0 \in \mathcal{R}^{n \times n} $. I want to generate other correlation matrices $C \in \mathcal{R}^{n \times n} $ that satisfies some linear constraints of the form
$$ A_i \textbf{.} C =b_i \,\,\, i= 1,2,3...m$$ where $A_i$ is a $n \times n $ symmetric constraint matrix. <$\textbf{.}$> implies the "element wise multiplication and sum" operator, i.e. $A_i \textbf{.} C= trace (A_i^TC)$. Also, $b_i$ is a scalar.
This is similar to a semidefinite programming problem except there is no cost function involved. Hence, every point in the feasible space (the space of all real semi-definite matrices of size $n \times n$ that satisfy the linear constraints) is a viable solution.
Also, my $n$ is around 500. So I am really looking for something that is very lean and scales well. What would be the best way to do this?
I have heard of CVX (cvx) that has semidefnite solvers. But I have never used it. What is a good way to set this up?
I deeply appreciate your help.