Upper bound for the weighted cosine Frobenius norm

Prove or disprove the conjecture that for every pair of orthogonal matrices U and V, the weighted cosine matrix satisfies \(\|C_\Sigma(U,V)\|_F^2\leq n^2/\gamma\), with equality attained at a common Hadamard frame whenever such a frame exists.

Background

The appendix studies the Frobenius norm of the matrix of singular-value-weighted cosines between rows of orthogonal frames. It proves that, for a single common frame, the identity frame achieves the minimum value and that an orthogonally scaled Hadamard frame is stationary when it exists. Numerical experiments suggest a corresponding global upper bound for arbitrary pairs of orthogonal frames, but the paper explicitly states that no proof is provided.

References

Theorem \ref{thm:min_max_weighted_cos_sym} does not establish that the Hadamard frame is a global maximizer. The supplied numerical experiments suggest the conjecture |C_\Sigma(U,V)|_F2\leq \frac{n2}{\gamma} with equality at a common Hadamard frame, but a proof is not provided here.

— A Geometric View of Adaptive Cross Approximation via Exterior Algebra  (2609.17947 - Loe et al., 16 Sep 2026) in Appendix A, Section “Weighted cosine and singular-vector coherence,” immediately after Theorem \ref{thm:min_max_weighted_cos_sym}