2026/02/28 by Isaac Dobes
Mathematics · #math.SP
39 pages
arxiv created 2026/07/30 · arxiv updated 2026/07/31
In this article we introduce higher-degree t-Hermitian forms, a tubal analogue of ordinary Hermitian forms of arbitrary degree. Through a synthesis of multilinear matrix multiplication and the t-product on third-order tensors, we show that t-Hermitian forms are in bijection with odd order tubal tensors satisfying certain symmetry conditions, which we call t-conjugate partial symmetry. After applying the Fast Fourier Transform along the tubal mode of the corresponding tubal tensor, t-Hermitian forms decompose into a family of classical Hermitian forms. This decomposition enables us to characterize positivity of t-Hermitian forms in terms of the spectra of the conjugate partially symmetric Fourier slices of its corresponding tubal tensor, yielding a tubal analogue of the spectral theorem for classical higher degree Hermitian forms. Then, as a central application of t-Hermitian forms, we study classical Hermitian forms induced by contractions along the tubal mode of t-conjugate partially symmetric tensors, characterizing when such contractions preserve positivity and deriving quantitative lower bounds for the positivity margin of the resulting classical Hermitian forms.