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Hi gals working in a quite crazy problem that our linear algebra teacher threwed at us out of nowhere, we have to give a proof of a formula for the spectral radius of these adjacency matrices, or in more pragmatic terms, the largest eigenvalue of matrices that follow this pattern, the matrices I need to study are the ones with $2^{n}-1 \times 2^{n}-1$ terms. that are like this.

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I know eigenvalues are nested square roots of two , and that the largest one i.e the radius will be one that has all positive signs for the nesting,also i know the determinant is zero and so the characteristic polynomial has no independent term, I have even build a pattern for the characteristic polynomial with a complex indexations and summations, but i am afraid that trying to compute the $2^{n}-1$ char poly, and the $2^{n}-3$ and using the division theorem won't really advance me at all in the proof, (maybe it will shoe me that they share eigenvalues or a sequence that is an eigenvalue but not that they are the biggest ones) the only hint the teacher gave us was to use induction, but i'm truly lost after giving it like 5 hours of work, is there any book that maybe touches and gives more hints about this not so basic linear algebra topic? maybe a very strong theorem on characteristic polynomials and the biggest eigenvalues?

Alejandro Quinche
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  • I don't see why the question was downvoted. The OP spent hours thinking about the problem, and the posted question included some of the OP's ideas and attempts. Also, it seems to me that the problem is somewhat challenging (at least I don't see an immediate resolution), so I think the post qualifies as a reasonable question for MSE. – quasi Jun 30 '19 at 01:30
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    You may search for "symmetric tridiagonal Toeplitz matrix" on the internet. There are known formulae for the eigenvalues and eigenvectors of such matrices. In your case, the eigenvalues are $2\cos\left(\frac{i\pi}{2^n}\right)$ for $i=1,2,\ldots,2^n-1$ and the spectral radius is $2\cos\left(\frac{\pi}{2^n}\right)$. – user1551 Jun 30 '19 at 09:17
  • Thanks, I don't mind being downvoted while there are people that can give me hints and I learn from it, I see the question is a little convoluted and being non-native that it isn't that very well written, hope to keep on improving! – Alejandro Quinche Jun 30 '19 at 17:00
  • , I will search @user15517 suggestion, I also found another paper mentioning the cosine connection that cites Chris Godsil and Gordon Royle. Algebraic Graph Theory I hope that I can follow the proof and present it, as I think it will follow from the recursive pattern of the polynomials I was looking before – Alejandro Quinche Jun 30 '19 at 17:07
  • @user1551 answer is correct but vague, i can't find any valuable paper for a sophomore in the subject, i can even find more info about non-simetric tridiagonals but all proofs i find sadly are quite complex and it seems my teacher suggestion about induction seems utterly useless for all of them – Alejandro Quinche Jul 01 '19 at 16:27
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    @AlejandroQuinche See How to find the eigenvalues of tridiagonal Toeplitz matrix? I am not sure what your teacher was hinting at, but I believe some answers in the aforementioned thread are elementary enough. – user1551 Jul 01 '19 at 17:39

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