Let $M$ be a an integer $s \times s$ matrix. Define the complexity of $M$ as $cx(M):= \inf \{ n \geq 0 | \exists C \in \mathbb{N} , \ \forall t \in \mathbb{N} : \| M^t \| \leq C t^{n-1} \}$. Here $\| M \|$ is some norm for matrices.
Question 1: Is such a definition of complexity of a matrix used/defined already somewhere?
For example a matrix with $M^k=id$ has complexity 1, the matrix \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} has complexity 2 and the matrix \begin{pmatrix} 2 & 2 \\ 0 & 1 \end{pmatrix} has infinite complexity.
For a finite connected poset $P$ with incidence matrix $M=(m_{i,j})$ such that $m_{i,j}=1$ if $i \leq j$ and $m_{i,j}=0$ else, define the Coxeter matrix of $P$ as $C=C_P:=-M^{-1} M^T$. Define the complexity $cx(P)$ of $P$ as the complexity of $C_P$. For example the complexity of the Boolean lattice is 1, while the complexity of the strong Bruhat order for $S_3$ is larger than 1.
Question 2: Is there a class of posets that shows that posets can have arbitrary complexity $ \geq 1$ or is the complexity bounded? Is the complexity of a poset related to some or bounded by some other well known invariants?
