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Let

$$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$

$K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right)$ and consider the sequence of matrices defined by $$ K_L = \left[H\otimes I_{2^{L-2}}\right]\left[I_2 \otimes K_{L-1}\right], $$ where $\otimes$ denotes the Kronecker product, and $I_n$ is the $n\times n$ identity matrix.

I am interested in the limiting behaviour of the singular values of $K_L$ as $L$ tends to infinity. Some calculation indicate that the $2^L\times 2^{L-1}$-matrix $K_L$ has $L$ non-zero singular values and that the empirical distribution of those non-zero singular values converges to some limit. Can this limit be described in terms of the matrix $H$?

I am wondering if it is possible to use some kind of fixed-point theorem to characterise the limit (in any sense) $\lim_{L\to\infty}K_L$ as an operator on some sequence space.


Edit: I did some more experiments and it seems that the limiting behaviour of the singular values of $K_L$ does not only depend on the matrix $H$, but also on the initial value $K_1$.

To illustrate this, let $K_1(\alpha)=\left(\begin{array}{c}1 \\ \alpha\end{array}\right)$ and consider the sequence $$ K_L(\alpha) = \left[H\otimes I_{2^{L-2}}\right]\left[I_2 \otimes K_{L-1}(\alpha)\right]. $$

The largest singular value of $K_{10}(\alpha)$ is depicted in the following figure. (The graph looks essentially the same for all $L\geq 4$ instead of $L=10$.)

The largest singular value of $K_{10}(\alpha)$

The minimum is approximately $(-.2936,0.7696)$.

This makes it unlikely for fixed-point arguments to work in this setting. I therefore modify my question and ask if the limiting behaviour of the singular values of $K_L$ (or $K_L(\alpha)$) can be characterisied directly in terms of $H$ and the initial value $K_1$ (or $K_1(\alpha)$).

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Could one propose a formulation in terms of tensors of an increasing rank over 2-dimensional vector space? My intuition is jammed with this Kronecker product stuff, but I’ll possibly understand something when saw tensors. –  Incnis Mrsi Aug 11 at 16:46
    
I do not think that I could contribute, but do the brackets in the definition of $K_L$ denote something special? Or are they just for emphasizing that the Kronecker product is evaluated before the matrix multiplication? –  flawr Nov 5 at 20:22

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