Jain: multiplicity two for absolute-cosine Hadamard powers

Tanvi Jain, Hadamard powers of some positive matrices, Linear Algebra and its Applications 528 (2017), 147–158, states in the final unnumbered paragraph of section 3, after the proof of Theorem 3.3:

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

For every natural m at least one, put n=2m+1 and C_ab=cos((a-b)pi/n), with a,b from zero through n-1. The conjecture concerns the matrix A(r) whose entries are |C_ab|^r for every real r at least zero. Every nonzero characteristic-polynomial root other than the top Fourier value must have algebraic multiplicity exactly two. The formal definition Defs.claim retains all these quantifiers and both exceptions.

The sentence is attributed to the publisher's indexed full text. The journal PDF was inaccessible, so its page number is unverified. The first version of arXiv:1803.06803 omits this sentence; the journal article is the source of the conjecture.

Reflection pairs the nonconstant Fourier modes of an odd real symmetric circulant matrix. This forces at least two occurrences of each representative, but does not exclude coincidences between different representatives. Jain's sign and inertia conclusions leave that separation question open.

Positive semidefiniteness, inertia, and the sign of an individual Fourier value do not separate two values of the same sign. The rank-two simplicity theorem for entrywise-positive positive semidefinite matrices does not apply: the cosine matrix has negative entries, and its entrywise absolute value need not have rank two. The truncated-cosine Jain–Karlin–Schoenberg kernel differs from the periodic absolute-cosine kernel here.

Put theta=pi/(2m+1), x_l=cos(l theta), and

$$ \lambda_j(r)=1+2\sum_{l=1}^m\cos(2jl\theta)x_l^r, \qquad P_j(r)=\prod_{s=0}^{j-1}(r-2s). $$

The bases satisfy 1>x_1>...>x_m>0. A generalized Descartes bound for real exponential sums, the exact trigonometric coefficient variation counts, and the forced zeros at 0,2,...,2j-2 give

$$ \operatorname{sgn}\lambda_j(r)=\operatorname{sgn}P_j(r),\qquad \operatorname{sgn}(\lambda_j(r)-\lambda_{j+d}(r)) =\operatorname{sgn}P_j(r),\quad d\in{1,2}. $$

The first identity has 1 at most j at most m. The second has j at least one, d equal to one or two, and j+d at most m. Both hold for every real r, including composite odd orders and zero intermediate gap coefficients. Steps one and two separate representatives within each sign class. Direct positivity of the cosine deficit separates the top mode. The real symmetric circulant factorization is

$$ \chi_{A(r)}(z)=(z-\lambda_0(r)) \prod_{j=1}^m(z-\lambda_j(r))^2. $$

Exactly one squared factor corresponds to a nonzero non-top root.

The analytic route to the spectral gap sign identities was derived with the assistance of the ChatGPT Pro model.

A counterexample would be an odd order at least three, a nonnegative real exponent, and a nonzero non-top characteristic-polynomial root whose multiplicity differs from two. A coincidence of distinct nonzero Fourier representatives would contradict the separation step. The zero-eigenvalue example below does not meet these conditions.

The formal development uses the exponential-sum Descartes theorem, trigonometric sign counts, forced even roots, exact gap signs, separation, and the real circulant characteristic-polynomial factorization. GapSigns proves both all-real sign identities; Separation proves the all-real mode separation statements; and Result proves the exact claim. The numerical tests below provide a separate finite check.

• PROVED (formalized): the exact sign formulas for every real r, strict top-mode separation and injectivity of nonzero Fourier representatives for every real r, and algebraic multiplicity exactly two for every nonzero non-top eigenvalue at every nonnegative real r.
• PROVED (formalized): the generalized Descartes bound and exact sign theorem for real exponential sums; the odd real symmetric circulant characteristic-polynomial factorization; the trigonometric coefficient variation counts and low-even-exponent forced zeros.
• PROVED (paper proof, not formalized as a separate theorem here): the exported claim restricts to r ≥ 0 as in the source; the formalized sign and separation theorems hold for every real r, and the same multiplicity argument applies verbatim for r < 0.
• PROVED (paper proof): the nonzero exception is necessary. At n=7 and r=2 the spectrum is 7/2, 7/4, 7/4, 0, 0, 0, 0, so the zero eigenvalue has multiplicity four.
• COMPUTED (mpmath, 40-digit arithmetic, outside the Lean development): sign formulas for m=1,...,15 and 67 exponents r in [-6,30], with zero violations. These are numerical checks and not part of the formal proof.
• OPEN: even orders and the analogous multiplicity statement for other entrywise functions of C.

The quotation and access limit are attributed to the journal source; the exact sentence page is unverified. The Jain note binds the DOI and article URL and distinguishes the journal and arXiv versions.

Searches covered the verbatim sentence, title, DOI, author and multiplicity keywords, MathDB, arXiv abstracts, and repository Problems, Library and D5 material. Relevant literature includes Jain's Hadamard powers of rank two, doubly nonnegative matrices (arXiv:2004.03909), Theorem 2(iv); Baslingker–Dan's work on Hadamard powers and positivity exponents (arXiv:2201.04443 and 2312.02766); Khare's Jain–Karlin–Schoenberg kernel treatment (arXiv:2008.05121) and lecture notes; Damm–Dietrich (arXiv:2207.08853); Belton–Guillot–Khare–Putinar (arXiv:1812.05482 and 2310.18041); and Alsaafin's thesis. The rank-two simplicity, positivity-exponent, truncated-kernel, and negativity-preservation results do not provide the spectral-gap separation required here.

No settlement of the conjecture was found in that scope. Citation indexes were unavailable, so the citing literature was not exhaustively checked; this is not a priority certification.