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

Isaac Dobes

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.

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

Abstract

We introduce -Hermitian forms of arbitrary degree , a natural extension of classical degree Hermitian forms obtained through a synthesis of the tensor transformation law and the -product of third-order tensors. We show that degree -Hermitian forms uniquely correspond to order Hermitian tensors arising as canonical representatives within the -algebra of Hermitian tensors equipped with the -product/-Einstein product. Applying the discrete Fourier transform, their corresponding -Hermitian forms decompose into collections of classical degree Hermitian forms. This decomposition yields a universal lifting property, allowing arbitrary collections of degree Hermitian forms to be viewed as a single structured object which preserves fundamental properties such as Hermitian positive-definiteness. For a distinguished class of -Hermitian forms, which we refer to as commutant -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.
Paper Structure (22 sections, 26 theorems, 210 equations, 3 algorithms)

This paper contains 22 sections, 26 theorems, 210 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor Unfolding
  4. Hermitian Forms
  5. Hypermatrix Representations of Hermitian Forms
  6. Positive Definite Tensors
  7. The t-product for Order 3 Tensors
  8. Definition/Basic Properties
  9. Tubewise t-Product
  10. t-Conjugate Transpose in Frequency Domain and Fourier Conjugation
  11. Degree $k$ t-Hermitian Forms
  12. Degree 1 t-Hermitian Forms
  13. Hermitian Positive Definite Degree 1 t-Hermitian Forms
  14. Arbitrary Degree t-Hermitian Forms
  15. The Universal Structure of Degree $k$ $t$-Hermitian Forms
  16. ...and 7 more sections

Key Result

Lemma 2.3

Suppose $\mathcal{A} = u_1\circ...\circ u_M\circ v_1\circ...\circ v_N\in \mathbb{C}^{m_1\times...\times m_M\times n_1\times...\times n_N}$. Then with $\mathcal{U} := u_1\circ...\circ u_M$ and $\mathcal{V} := v_1\circ...\circ v_N$.

Theorems & Definitions (96)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 86 more