bibkey: jain2017hadamard
authors: Tanvi Jain
year: 2017
title: "Hadamard powers of some positive matrices"
doi: 10.1016/j.laa.2016.06.030
url: https://doi.org/10.1016/j.laa.2016.06.030
claim: "The final paragraph of section 3 conjectures multiplicity two for every nonzero non-top eigenvalue of odd absolute-cosine Hadamard powers."
strata_touched:
  - D5/S3/Quantum/Matrix/JainAbsCosine/Defs
  - D5/S3/Quantum/Matrix/JainAbsCosine/Result
license: citation-only
triage: anchor

Absolute-cosine Hadamard powers

Locator

DOI: 10.1016/j.laa.2016.06.030

Article: https://doi.org/10.1016/j.laa.2016.06.030

Tanvi Jain, Hadamard powers of some positive matrices, Linear Algebra and its Applications 528 (2017), 147–158. The conjecture occurs in the last unnumbered paragraph of section 3, immediately after the proof of Theorem 3.3. Zbl 1398.15038; MR 3652842.

Related author manuscript: https://arxiv.org/abs/1803.06803. The first arXiv version omits the conjecture sentence; the journal version is the source of the quotation below.

Source statement

We conjecture that the multiplicity of every nonzero eigenvalue of |C|∘r∘ (except the top eigenvalue) equals two.

Here the order is odd, n=2m+1 at least three, r is real and nonnegative, and C has entries cos((a-b)pi/n). The notation denotes entrywise absolute value followed by the entrywise real power. Proposition 3.2 and Theorem 3.3 provide Fourier and inertia information; equal signs of distinct Fourier representatives do not imply their equality or separation.

Source boundary

The quotation is attributed to the publisher’s indexed full text. The journal PDF was inaccessible, so the exact page number of the sentence is unverified. The DOI and article URL identify the journal source; the quotation is not attributed to the arXiv manuscript. Bibliographic and source-access evidence does not constitute an exhaustive citation-index check or a priority certification.