I try to analyze the following iterative matrix product:
$x_{i+1} = A_{i} \cdot x_{i}$.
The matrices $A_i$ are defined as follows:
$A_i = \begin{bmatrix} \frac{1}{2\sqrt{x_{i,1}^2 + y_{i,1}^2}} & \frac{1}{2} & 0 & 0 & \dots & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & \dots & 0 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & \dots & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 &0 & \frac{1}{2} & \frac{1}{2\sqrt{x_{i,n}^2 + y_{i,n} ^2}} \end{bmatrix}$
Note that all rows except of the first and the last row are stochastic.
The initial matrix $x_1$ looks like this. A requirement is that the length of each row vector is at most $1$. $x_1 = \begin{bmatrix} x_{1,1} & y_{1,1} \\ x_{1,2} & y_{1,2} \\ \vdots & \vdots \\ x_{1,n} &y_{1,n} \end{bmatrix}$.
In case we assume that all row vectors are initially different, it turns out that the product converges to a vector $x_{lim}$ in which every row vector has length $1$. Additionally, all row vectors are identical then.
However, I want to theoretically analyze this behavior.
Does anyone of you have an idea of how to analyze this product? I tried to analyze it via contraction coefficients. With these, I can only that the products of matrices converge to a rank 1 matrix, but I do not have an idea about the number of required iterations.