[warning: biologist asking math question]
In a linear dynamical system, what feature of the matrix controls the speed of the trajectory in state space?
Say I have a matrix M describing how the system evolves per discrete time unit t: $$ \mathbf{x}_{t+1} = \mathbf{M} \cdot \mathbf{x}_t $$
The state of the system n time steps later is given by: $$ \mathbf{x}_{t+n} = \mathbf{M}^{n} \cdot \mathbf{x}_t $$
How should I modify M so as to reach state $\mathbf{x}_{t+n}$ in arbitrary fewer (or more) time steps? In other words, what is matrix P so that $\mathbf{P}^{k \cdot n} = \mathbf{M}^{n}$ for arbitrary $k$ ? Is it a trivial question?
thanks,
It seems that you want to compute a $k$th root of $M$.
If this is so, it really depends on what entries you allow. But at least you can do everything easily in the following (much simpler) situation: $M$ is diagonalizable and all eigenvalues of $M$ are positive real numbers.
Then you have $S^{-1}MS=D$ for some change of basis matrix $S$ and some diagonal matrix $D$ having the eigenvalues of $M$ on its main diagonal. Simply define $D'$ to be the diagonal matrix having entries on its main diagonal equal to the $k$th roots of the entries on the main diagonal of $D$ (this choice is not unique). Then the matrix $P=SD'S^{-1}$ satisfies $P^k=M$.