I have a "simple" linear algebra problem that popped up as part of a different problem I was working on.
For given $\ell_A,\ell_B$, I fix a sequence $(\ell_n)_{n=1}^{\infty}\subset\{\ell_A,\ell_B\}$. I then define a sequence of linear maps on $\mathbb R^2$: $$T_nv=\left(\prod_{j=1}^nA_j\right)v,$$ $$A_j=\begin{bmatrix}(1+\frac{1}{2}\tanh(\sqrt{-z}\ell_j))e^{-\sqrt{-z}L}&\frac{1}{2}\tanh(\sqrt{-z}\ell_j)e^{-\sqrt{-z}L}\\-\frac{1}{2}\tanh(\sqrt{-z}\ell_j)e^{\sqrt{-z}L}&(1-\frac{1}{2}\tanh(\sqrt{-z}\ell_j))e^{\sqrt{-z}L}\end{bmatrix} $$ where $z\in\mathbb C$ and $L>0$ are given constants.
Bottom line, I act one a given vector $v$ with a sequence of (invertible) matrices $A_n$, giving me a discrete linear dynamical system on $\mathbb R^2$. This dynamical systems is "relatively" simple, in the sense that in each step I only act with one of two possible choices of matrices (depending on the value of $\ell_j$).
My particular interest is computing the image of the linear subspace $sp\{(0,1)^t\}$ (the $y$ axis) under the iterations of this dynamical system. This is equivalent to computing the action of the system on $(0,1)^t$ and taking the linear span of this image (all of its scalar multiples).
If the sequence $(\ell_n)$ is constant (so always acting with the same matrix), this is a matter of diagonalization of the matrix $A_j$, because I basically want to compute its powers and act with them on the initial condition $(0,1)$. But sadly, the sequence is not constant (not even periodic), and so this approach does not work (note that the two possible choices for $A_j$ do not commute).
I am wondering how to go about this. Perhaps something about the matrices $A_j$ can make the computation easier, but I don't see anything at this point. The difficulty comes from the fact that I want to precisely know this image, not just study it qualitively, so I need to really compute the image.
Another question, which should probably be easier, is the following - I conjecture that for some values of $z$, the image of this subspace converges to some fixed subspace (in the usual topology) as $n\rightarrow \infty$. I would like to show this convergence and compute the associated limiting subspace. Does anyone have an idea on how to do this? Again, if the sequence is constant then this is relatively simple, but my sequence $(\ell_n)_{n=1}^{\infty}$ is more general.
Thanks for your help. If anyone wants more information about the sequence $(\ell_n)$ feel free to ask, although it's a bit technical so I'm hoping for a more general method.
This isn't a complete answer, but I think it's a good start. Consider the eigendecomposition $A_{\ell_A} = Q_A\Lambda_AQ_A^H$ and $A_{\ell_B} = Q_B\Lambda_BQ_B^H$. Let $$\{\alpha_m^{\beta_m}\} = \text{red} \{\ell_m\},$$ where red is the reduction function, e.g. $\text{red}\{\ell_A,\ell_B,\ell_B,\ell_A\} = \{\ell_A, \ell_B^2, \ell_A\}$. From here, we just need to characterize how $Q_A^HQ_B$ and $Q_B^HQ_A$ change the system. Obviously, once we know one, we know the other. Here's the result I got using wolframalpha:
There's not really a simple form. But you can at least work from there. To exam it further, you probably want to look for conditions of $A_A$ and $A_B$ being unitarily similar, i.e. how do the different values affect the eigendecomposition. The alternative is to explore how the eigenvalues change the mapping of $Q_A^HQ_B$ as a function of $z,\ell_A,\ell_B$.