<s>[INST] Finding matrices $A$ such that the entries of $A^n$ have specified signs What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of the matrix power $A^n$. Under what conditions on $A$ does the following hold:
$$
\text{sgn}\ (A^n)_{i,j}=(-1)^{j+1}
$$
 [/INST] To follow up on Terry's comment (actually, I did the computation before seeing it, but whatever): in the parabolic case, where the matrix has the form
$$A = \begin{pmatrix} d & -b \\ -c & a\end{pmatrix} \begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & b \\ c & d\end{pmatrix},$$
Then 
$$A^n = \begin{pmatrix}1 + c d n x & d^2 n x\\ - c^2 n x & 1 - c d n x\end{pmatrix}.$$
So, if $c \in \mathbb{R},$ then $x < 0$ (from the bottom left), and so $c < 0$ from the bottom right, and we are fine (that is, if $|c d x| > 1,$ then any $a, b$ works). Since $x$ is real, it is clear from the bottom left that $c$ is also, so this finishes the parabolic case.
In the hyperbolic case,
$$A = \begin{pmatrix} d & -b \\ -c & a\end{pmatrix} \begin{pmatrix} x & 0 \\ 0 & 1/x\end{pmatrix} \begin{pmatrix} a & b \\ c & d\end{pmatrix},$$ where $x>1.$
and $$A^n = \begin{pmatrix}=b c x^{-n} + a d x^n & b d x^{-n}(-1 + x^{2 n})\\ a c(x^{-n}+ x^n) & ad x^{-n} - b c x^n\end{pmatrix}.$$ For $n \gg 1,$ we have
$$A^n \sim x^n \begin{pmatrix} ad & b d\\ -ac & -bc\end{pmatrix},$$ which implies that either $a < 0, b>0, c> 0, d<0,$ or $a > 0, b< 0, c<0, d> 0.$ Neither of which is compatible with the base case $n=1$ satisfying your condition.
In the elliptic case, the eigenvalues are either roots of unity, so $A^n = I,$ infinitely often, or not, in which case $A^n$ is very close to $I$ infinitely often, so either way, your condition fails.
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