# Limit of sequence of growing matrices

Let

$$H=(01/201/21/201/201/2001/201/21/20), H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right),$$

$$K1=(10)K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right)$$ and consider the sequence of matrices defined by
$$KL=[H⊗I2L−2]⏟2L×2L[I2⊗KL−1]⏟2L×2L−1∈R2L×2L−1, K_L = \underset{2^{L}\times 2^{L}}{\underbrace{\left[H\otimes I_{2^{L-2}}\right]}}\underset{2^{L}\times 2^{L-1}}{\underbrace{\left[I_2 \otimes K_{L-1}\right]}}\in\mathbb{R}^{2^L\times 2^{L-1}},$$
where $$⊗\otimes$$ denotes the Kronecker product, and $$InI_n$$ is the $$n×nn\times n$$ identity matrix.

I am interested in the limiting behaviour of the singular values of $$KLK_L$$ as $$LL$$ tends to infinity. Some calculations indicate that the $$2L×2L−12^L\times 2^{L-1}$$-matrix $$KLK_L$$ has $$LL$$ non-zero singular values and that the empirical distribution of those nonzero singular values converges to some limit. Can this limit be described in terms of the matrix $$HH$$?

I am wondering if it is possible to use some kind of fixed-point theorem to characterise the limit (in any sense) $$limL→∞KL\lim_{L\to\infty}K_L$$ 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 $$KLK_L$$ does not only depend on the matrix $$HH$$, but also on the initial value $$K1K_1$$.

To illustrate this, let $$K1(α)=(1α)K_1(\alpha)=\left(\begin{array}{c}1 \\ \alpha\end{array}\right)$$ and consider the sequence
$$KL(α)=[H⊗I2L−2][I2⊗KL−1(α)]. K_L(\alpha) = \left[H\otimes I_{2^{L-2}}\right]\left[I_2 \otimes K_{L-1}(\alpha)\right].$$

The largest singular value of $$K10(α)K_{10}(\alpha)$$ is depicted in the following figure. (The graph looks essentially the same for all $$L≥4L\geq 4$$ instead of $$L=10L=10$$.)

$K_{10}(\alpha)$” />

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

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 $$KLK_L$$ (or $$KL(α)K_L(\alpha)$$) can be characterised directly in terms of $$HH$$ and the initial value $$K1K_1$$ (or $$K1(α)K_1(\alpha)$$).

Edit 2 (March 2015):
As the question is still receiving attention, let me add that I came up with a conjecture for the asymptotic behaviour of the singular values of $$KL(α)K_L(\alpha)$$, as detailed in this MO post.