t-Hermitian Forms of Arbitrary Degree, Their Spectral Structure, and Positivity

TL;DR

This paper introduces a new class of t-Hermitian forms of arbitrary degree, explores their spectral properties, and establishes a framework connecting tensor eigenvalues and positivity, extending classical Hermitian theory.

Contribution

It defines t-Hermitian forms of any degree, relates them to Hermitian tensors via a novel algebraic structure, and develops a spectral theory linking tensor and matrix eigenvalues to positivity.

Findings

t-Hermitian forms correspond to Hermitian tensors with a unique algebraic structure

Decomposition via Fourier transform relates t-Hermitian forms to classical Hermitian forms

Tensor eigenvalues characterize Hermitian positive-definiteness, matrix eigenvalues imply positivity

Abstract

We introduce $t$-Hermitian forms of arbitrary degree $k$, a natural extension of classical degree $k$ Hermitian forms obtained through a synthesis of the tensor transformation law and the $t$-product of third-order tensors. We show that degree $k$ $t$-Hermitian forms uniquely correspond to order $2k+1$ Hermitian tensors arising as canonical representatives within the $^*$-algebra of Hermitian tensors equipped with the $t$-product/$t$-Einstein product. Applying the discrete Fourier transform, their corresponding $t$-Hermitian forms decompose into collections of classical degree $k$ Hermitian forms. This decomposition yields a universal lifting property, allowing arbitrary collections of degree $k$ Hermitian forms to be viewed as a single structured object which preserves fundamental properties such as Hermitian positive-definiteness. For a distinguished class of $t$-Hermitian forms, which we refer to as commutant $t$-Hermitian forms, we establish a spectral theory extending the classical spectral characterization of Hermitian tensors. In particular, we relate Hermitian positive-definiteness to two distinct classes of eigenvalues: the tensor eigenvalues of each frontal slice of the form's corresponding tensor, and matrix eigenvalues obtained after unfolding each of these slices into matrices. Indeed, we prove that the tensor eigenvalues characterize Hermitian positive-definiteness; moreover, positivity of the matrix eigenvalues implies Hermitian positive-definiteness, however the converse fails in general. Together, these results provide a unified framework for structured interactions among higher-degree Hermitian forms and lay the groundwork for further study, especially with regards to the consequences of the spectral hierarchy of positivity inherent in this setting.