Here's a simple but potential research problem that I am learning about.

Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and given the number "1" or inactive and given the number "0".

So, the system has $2^N$ possible states.

Now, if I am able to compute and find the state-transition-matrix of this system, it should be a $2^N \times 2^N$ matrix.

(Computability of the matrix is a different question.)

But what I am confused about is this: if I am understanding this problem correctly, the state-transition-matrix is a generalization of the transition matrix of, say, an Ergodic Markov Chain - a matrix that most students learn from basic probability theory (the entries are nonnegative and represent probabilities). Applying the transition matrix to a probability vector "updates" the probability vector that describes the Markov Chain.

But, is the same thing true of the more general state-transition-matrix of the physical system -- a system that is not necessarily a Markov Chain?

I.e., if I can compute this state-transition-matrix, would applying this matrix to some vector then "update" the vector that describes the current state of the system?

If the answer to the above is "yes", then I see one technical issue:

the matrix multiplication wouldn't make sense, as I would be applying a $2^N \times 2^N$ matrix to an **N-tuple vector**, with each component taking on the values 1 or 0. The vector wouldn't be of length $2^N$.

Where am I going wrong here?

Thanks in advance,