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Suppose I have a matrix where each successive row is a left-shift of the previous row (with a new value coming in on the right), e.g.,

$$A = \begin{bmatrix} 1 & 2\\ 2&3\\ 3&4 \end{bmatrix}$$

Observe that each square subset of this matrix is symmetric. I was wondering if for a general $m \times n$, where $m \gg n$, matrix $A$ with this shifting property we could prove a lower bound on the minimum singular value that is dependent on $m$, such that all the values are roughly similar with high probability, say they are normally distributed.

I've looked at lower bounds but they often restrict to the minimum singular value of a square matrix contained within $A$, such as Singular value inequality for block matrices

I've done some simulations and I see that the the minimum singular value increases with the number of rows $m$, which I expected, given Changes in singular values of matrix when rows are added. (this link is exactly what I want, but if only it were some sort of strict inequality I could prove WHP) Is there any sort of theoretical way to prove this? Any help would be appreciated, even just recommendations on what textbooks to look at, since this kind of linear algebra material wasn't covered in any of the undergrad courses I took.

Rodrigo de Azevedo
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Will
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  • Your matrix is a submatrix of a Hankel matrix – Rodrigo de Azevedo May 27 '23 at 06:16
  • Thank you very much, did not know there was a name for it! – Will May 27 '23 at 18:02
  • After permuting the columns, is it also a sub matrix of a circulant matrix? – Rodrigo de Azevedo May 27 '23 at 18:13
  • I suppose we could say it is a submatrix of a circulant matrix (after flipping the column order), but a massive $m \times m$ one. I'm not exactly sure I understand what you're hinting at. Also, by looking at these larger matrices, since I'm trying to achieve lower bound on the smallest singular value, wouldn't these give an upperbound instead? – Will May 27 '23 at 19:44

1 Answers1

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The topic is specific and there is no exact answer, but I would like to suggest a number of actions that will help you

  1. research in the area of Gankele matrices and singular values of such matrices. Hankel matrices have properties that may be useful in your question.
  2. study the properties of singular values and their behaviour when rows or columns are added to a matrix. This can help to understand how singular values change as the number of rows in your matrix increases.
  3. it is also worth noting the statistical properties of the singular values of random matrices, in particular how they are distributed. This can give you some idea of how your values might be distributed, and how high the probability is that they will be about the same.

These are resources that can help: 1, 2, 3

Partim
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  • To the second point, the link I put in the post, it shows that the singular values are nondecreasing as the number of rows increase. I guess something from the first and third point will help me make it strict? – Will May 28 '23 at 01:40