What is the relation between states in the original and reduced model

Hello, I'm using balred to simplify a state space model. This works fine, but I want to use this to simulate and compare it with the original model results, and this from specific inital conditions. The problem is that I don't know how to convert the initial state in the original model to the states in the reduced model. I would think that there should be a conversion matrix (or maybe 2?).

So:

dX/dt = A.X + B.U

 --> reduced: dZ/dt = Ar.Z + Br.U

The question: is there something like a matrix Q?

   Z0 = Q.X0 (or another relationship)