I am trying to build an autoregressive model such that $\mathbf{W}^t = \mathbf{W}^{t-1}\mathbf{M}$ where $\mathbf{M} \in \mathbb{R^{k \times k}}$ and $\mathbf{W} \in \mathbb{R^{p \times k}}$ where $p > k$. Is there a constraint to put on matrix $\mathbf{M}$ such that $\mathbf{W}$ smoothly evolves?
Edit 1: Here smoothly means that difference between each element of $\mathbf{W}^{t-1}$ and $\mathbf{W}^{t}$ is not very large or we can say that the difference is bounded by some term $\alpha$.
I would suggest the following constraint: All eigenvalues of $M$ must have absolute values less than $ 1$.
Observe that $W^t=WM^{t-1}$.
Define a matrix norm $\|\|$ as follows: Let $\|M\|=\max_{i,j}|[M]_{i,j}|$ be the absolute value of the element of $WM$ whose absolute value is greatest.
On this Wikipedia page about Spectral radius, it is proved that , for any matrix norm $\|\cdot\|$, $\|M^t\|$ tends to zero if and only if all eigenvalues of $M$ have absolute value less than $1$. The same page also proves that $\|M^t\|$ tends to infinity when $M$ has an eigenvalue with absolute value greater than $1$.
If $M^t$ is bounded then $W^t$ is bounded and, since $W^t-W^{t-1}=W^{t-1}(M-I)$, the difference $W_{t+1}-W_{t}$ is also bounded.