Let

H=(01/201/21/201/201/2001/201/21/20),

K1=(10) and consider the sequence of matrices defined by

KL=[H⊗I2L−2]⏟2L×2L[I2⊗KL−1]⏟2L×2L−1∈R2L×2L−1,

where ⊗ denotes the Kronecker product, and In is the n×n identity matrix.I am interested in the limiting behaviour of the singular values of KL as L tends to infinity. Some calculations indicate that the 2L×2L−1-matrix KL 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) limL→∞KL 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 KL does not only depend on the matrix H, but also on the initial value K1.To illustrate this, let K1(α)=(1α) and consider the sequence

KL(α)=[H⊗I2L−2][I2⊗KL−1(α)].The largest singular value of K10(α) is depicted in the following figure. (The graph looks essentially the same for all L≥4 instead of L=10.)

K10(α)” />

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 KL (or KL(α)) can be characterised directly in terms of H and the initial value K1 (or K1(α)).

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), as detailed in this MO post.

**Answer**

**Attribution***Source : Link , Question Author : Eckhard , Answer Author : Community*