Speed of linear dynamical system trajectory

Question

[warning: biologist asking a math question]

In a linear dynamical system (LDS), what feature of the matrix controls the speed of the trajectory in state space?

Say I have a matrix M describing how the LDS evolves per discrete time unit t. State after 10 t is given by M^10, and I'll call it the final state. For the same initial condition, how should I modify M to make it reach the final state in arbitrary fewer (or more) time steps? Is it trivial?

thanks,