# Limit of sequence of growing matrices

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 $$K_L = \underset{2^{L}\times 2^{L}}{\underbrace{\left[H\otimes I_{2^{L-2}}\right]}}\underset{2^{L}\times 2^{L-1}}{\underbrace{\left[I_2 \otimes K_{L-1}\right]}}\in\mathbb{R}^{2^L\times 2^{L-1}},$$ where $$\otimes$$ denotes the Kronecker product, and $$I_n$$ 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 calculations 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 nonzero 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 $$L=10$$.) $K_{10}(\alpha)$" />

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 characterised directly in terms of $$H$$ and the initial value $$K_1$$ (or $$K_1(\alpha)$$).

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 $$K_L(\alpha)$$, as detailed in this MO post.

• 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 '14 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 '14 at 20:22
• @flawr It's only that Kronecker product is evaluated before the matrix multiplication – Paul Jan 25 '16 at 16:32
• This has been answered on MathOverflow, should it be deleted here then? – Ramanujan Jan 8 '19 at 8:19
• What was the motivation for this problem? – Vítězslav Štembera May 1 '19 at 16:47