<s>[INST] Possible growth rates of a matrix entry with respect to exponentiation Let $A = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$, so $A^n = \begin{pmatrix}1 & n \\ 0 & 1\end{pmatrix}$. Thus, $(A^n)_{1,1} = 1 = \Theta(1)$, and $(A^n)_{1,2} = n = \Theta(n)$.
Given a constant $c$, if $B = \begin{pmatrix} c/2 & c/2 \\ c/2 & c/2\end{pmatrix}$ then $B^n = \begin{pmatrix} c^n/2 & c^n/2 \\ c^n/2 & c^n/2\end{pmatrix}$. Thus, $(B^n)_{1,1} = c^n/2 = \Theta(c^n)$.
We have observed three possible growth rates of a particular entry with respect to the exponent of a matrix, constant, linear, and exponential. What other growth rates are possible? For example, does there exist an $m\times m$ matrix $C$ and indices $i,j$ such that $(C^n)_{i,j} = \Theta(n^2)$?
 [/INST] Consider $A=I+S,$ where $S$ a matrix with entries $s_{i,i+1}=1$ and $0$ otherwise. If the dimension is equal $m,$ then $$A^n=I+ \sum_{k=1}^{m-1} {n\choose k}S^k$$ Thus $$(A^n)_{1,1+j}={n\choose j}\approx {n^j\over j!}\quad \qquad 1\le j\le m-1$$
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