Source-linked AI summary
E-Determinants of Tensors
Shenglong Hu, Zheng-Hai Huang, Chen Ling, Liqun Qi
TL;DR
The paper asks how determinant-like and eigenvalue-related theory can be extended from symmetric hyperdeterminants and matrices to general tensors. It defines the E-determinant via polynomial-system resultants, proves determinant analogues across tensor operations and inequalities, and develops characteristic-polynomial results including explicit two-dimensional formulas. These results also yield a solvability guarantee for systems with triangular leading tensors having nonzero diagonal entries.
Problem
The paper studies how the resultant associated with a general tensor can support determinant-like properties, polynomial-system solvability, and tensor eigenvalue theory beyond the symmetric-tensor setting.
Method
It defines Edet(T) as the resultant of T x^(m−1) = 0 and analyzes its composition, block-tensor, characteristic-polynomial, trace, inequality, and solvability properties.
Results
The paper establishes determinant analogues for general tensors, shows solvability for systems whose leading tensor is triangular with nonzero diagonal entries, and gives explicit E-determinant and characteristic-polynomial formulas in dimension two.
Takeaways & Limitations
The E-determinant provides a framework linking general tensor algebra, polynomial-system solvability, determinant inequalities, and tensor characteristic polynomials.
Takeaways & Limitations
The paper leaves open whether Conjecture 4.1 holds in general and identifies further issues for future research; its general assumptions take m, n ≥ 2 and complex-valued tensors.
Abstract
from arXiv · showhide
We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard's inequality, Gersgrin's inequality and Minikowski's inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.
1 Introduction
The paper defines the E-determinant for general tensors through polynomial-system resultants and develops determinant-like properties connected to tensor eigenvalues. It applies this theory to polynomial solvability, tensor composition and block structure, characteristic polynomials, inequalities, and two-dimensional formulas.
- Definition and motivation: The E-determinant of a tensor is defined as the resultant of the polynomial system T x^(m−1) = 0.This extends the symmetric hyperdeterminant framework from symmetric tensors to general tensors.
- Polynomial systems: The theory characterizes solvability of polynomial systems through the E-determinant of their leading coefficient tensor.A triangular leading coefficient tensor with nonzero diagonal elements guarantees that the polynomial system has a solution.
- Tensor operations: Composition of homogeneous tensor-induced polynomial maps yields a tensor W of order (p−1)(q−1)+1, with Edet(W) = 0 exactly when Edet(U)Edet(V) = 0.The paper also notes that the related conjecture is proved when min{p, q} = 2.
- Tensor operations: For tensors with an upper triangular block structure, the paper gives an E-determinant expression based on the E-determinants of the two diagonal subtensors.This generalizes the corresponding block factorization property of matrix determinants.
- Characteristic polynomials: The E-determinant supports characteristic-polynomial trace formulas, eigenvalue representations, and a computable positive-semidefiniteness condition under Assumption 8.1.Explicit formulas for the E-determinant and characteristic-polynomial coefficients are given when the dimension is two.
- Inequalities: The paper generalizes Hadamard’s, Gershgorin’s, and Minkowski’s inequalities to estimate tensor E-determinants from tensor entries.These results extend several determinant inequalities from matrices to tensors.
2 Basic Properties of the E-Determinant
This section defines the tensor characteristic polynomial through the E-determinant and establishes foundational algebraic properties of the E-determinant, including homogeneity, irreducibility, normalization, and a zero criterion.
- Characteristic polynomial: The characteristic polynomial of tensor T is ψ(λ) = Edet(λE − T), with roots corresponding to tensor eigenvalues.A root’s multiplicity is defined as the algebraic multiplicity of the associated eigenvalue.
- Characteristic polynomial: The characteristic polynomial ψ is homogeneous of degree n(m − 1)^(n−1).This degree matches the total degree established for the E-determinant in the tensor entries.
- Algebraic properties: The E-determinant is homogeneous of degree (m − 1)^(n−1) in each diagonal-variable group and degree n(m − 1)^(n−1) overall.These degree properties describe how the resultant depends on tensor entries.
- Algebraic properties: Edet(T) is irreducible and homogeneous in the tensor entries, while the identity tensor satisfies Edet(E) = 1.The identity-tensor normalization parallels a basic determinant property.
- Zero criterion: If all entries associated with one output index i vanish, then Edet(T) = 0; in particular, the zero tensor has zero E-determinant.This follows from the E-determinant’s homogeneity in the corresponding diagonal-variable group.
3 Solvability of Polynomial Equations
The E-determinant extends determinant-style solvability criteria to homogeneous and nonhomogeneous polynomial systems. Its nonzero value guarantees solvability for every right-hand side and lower-degree polynomial perturbation.
- Det(A) ≠ 0 characterizes a unique solution to Ax = b, motivating the analogous E-determinant criterion for polynomial systems.
- Edet(T) = 0 if and only if T x^(m−1) = 0 has a nonzero solution.
- Edet(T) ≠ 0 implies solvability of T x^(m−1) = B_(m−1)x^(m−2) + ··· + B_3x^2 + Ax + b for every b, A, and B_j of the stated orders.
- The proof embeds the polynomial system into a homogeneous tensor system in one higher dimension and derives a nonzero solution from Edet(T) ≠ 0.
- E-determinants therefore serve as criteria for solvability of nonlinear polynomial equations, paralleling determinants for linear equations.
4 Composition of Homogenous Polynomial Maps
The paper defines tensor composition through composition of the induced homogeneous polynomial maps and proves an E-determinant zero-product law. A broader multiplicative conjecture is established when one tensor is matrix-order.
- A tensor induces a homogeneous polynomial map, and composing tensors of orders p and q yields order 1 + (p − 1)(q − 1).
- The coefficient construction in equation (5) makes tensor composition uniquely defined when orders exceed two, unlike the matrix case.
- Edet(U ◦ V) = 0 if and only if Edet(U)Edet(V) = 0.
- Consequently, zero is an eigenvalue of U ◦ V exactly when zero is an eigenvalue of U or V.
- The conjectured full product formula is proved when p = 2 or q = 2, extending the matrix Cauchy–Binet case.
5 Block Tensors
The block-tensor result generalizes determinant factorization for block-triangular matrices. Under a specified zero-pattern condition, the E-determinant vanishes exactly when one diagonal sub-tensor has vanishing E-determinant.
- For block-triangular matrices, determinant factorization motivates an analogous tensor product formula.
- A sub-tensor is formed by restricting every tensor index to a selected index set.
- If the specified entries coupling the lower block to the first block vanish, T has diagonal sub-tensors U and V satisfying the block condition.
- Edet(T) = 0 implies Edet(U)Edet(V) = 0, and the converse is proved by constructing a nonzero vector from a null vector of either sub-tensor.
- The proof uses the block equations and the irreducibility and homogeneity of the E-determinant to establish the product relation.
6 A Simple Application: Triangular Tensors
For triangular tensors, the characteristic spectrum is determined by diagonal entries, and each diagonal entry has multiplicity (m − 1)^(n − 1). This yields solvability for broad polynomial systems with nonzero triangular leading coefficients.
- An upper triangular tensor has t_i1…im = 0 when min{i_2,…,i_m} < i_1, while a lower triangular tensor vanishes when max{i_2,…,i_m} > i_1.
- For a triangular tensor, the spectrum is σ(T) = {t_i…i | i = 1,…,n}.
- Each diagonal entry t_i…i has algebraic multiplicity (m − 1)^(n − 1).
- If the leading tensor is triangular with nonzero diagonal elements, the associated polynomial system has a solution in C^n for every permitted b, A, and B_j.
- The composition result is established for two upper triangular tensors or two lower triangular tensors, but the paper does not provide details.
7 A Trace Formula of the E-Determinant
The paper develops a trace formula for the E-determinant using differential operators and Schur-polynomial expansions. The resulting finite formula expresses the E-determinant through tensor traces.
- Trace formula: A trace formula for the E-determinant is derived from a result on resultants of homogeneous polynomial systems.The construction uses differential operators and extends a determinant-related trace identity.
- Trace formula: Schur-polynomial expansions connect the traces Tr_1(T), ..., Tr_k(T) to the E-determinant of E − T.The expansion is first obtained in an infinite form and then converted into a finite expression.
- Trace formula: The d-th trace Tr_d(T) is introduced through differential operators associated with an auxiliary matrix.These operators belong to the algebra generated by partial derivatives of the auxiliary matrix variables.
- Trace formula: Each Tr_d(T) is a homogeneous polynomial of degree d in the tensor entries.The homogeneity supports the finite truncation of the E-determinant expansion.
- Trace formula: The resulting finite trace formula involves the differential operators and is used later to obtain an explicit formula when n = 2.The paper states that the two-dimensional specialization is developed in Section 9.
8 The Characteristic Polynomial
The characteristic polynomial is defined through the E-determinant, allowing the paper to relate tensor eigenvalues, traces, and positive semidefiniteness. Under stated assumptions, coefficient conditions characterize positive semidefiniteness in important cases.
- Characteristic polynomial: The characteristic polynomial of T is ψ(λ) = Edet(λE − T), and the paper derives both a trace formula and an eigenvalue representation for it.This connects the E-determinant directly to tensor spectral information.
- Positive semidefiniteness: Under Assumption 8.1, a symmetric tensor with even order is positive semidefinite exactly when all its real eigenvalues are nonnegative.Assumption 8.1 requires every negative eigenvalue to have an associated real eigenvector.
- Positive semidefiniteness: The coefficients of the characteristic polynomial provide conditions for positive semidefiniteness under the paper’s stated parity and eigenvalue assumptions.For even dimension, nonnegative real parts of all complex eigenvalues are additionally required for the equivalence.
- Positive semidefiniteness: The coefficient test can avoid computing all real eigenvalues, including the smallest one, when the additional hypotheses hold.The paper identifies this as the practical significance of Proposition 8.1.
- Characteristic polynomial: The solvability of homogeneous polynomial equations is characterized by whether the underlying tensor has a zero eigenvalue.This conclusion follows from the relationship between the characteristic polynomial and the E-determinant.
9 Explicit Formulae When n = 2
For dimension n = 2, the paper develops combinatorial trace expansions that yield explicit characteristic-polynomial coefficients and an explicit E-determinant formula. It also gives an explicit expression for Tr_2(T) for arbitrary order.
- Section overview: The characteristic polynomial and E-determinant depend on traces through d = 1, ..., n(m − 1)^(n−1), but computing these traces is complicated.The section therefore develops preliminary trace-computation results.
- Trace expansions: For arbitrary order and dimension, the section gives an explicit formula for Tr_2(T).The displayed expression combines diagonal and off-diagonal tensor entries.
- Trace expansions: A combinatorial counting lemma enumerates terms in trace expansions involving diagonal and off-diagonal tensor entries.The count is organized using packaged elements and compositions indexed by bounded integer tuples.
- Trace expansions: Lemmas 9.1 and 9.3 provide the trace identities needed to compute characteristic-polynomial coefficients when n = 2.The identities use the auxiliary sets D_s and E_s defined from bounded compositions.
- Explicit formulae: For n = 2, the coefficients of the characteristic polynomial are given explicitly in terms of the entries of the underlying tensor.The paper presents this as an alternative to Sylvester’s formula.
- Explicit formulae: For n = 2, the E-determinant is obtained explicitly from the characteristic polynomial at λ = 0.The relation used is Edet(T) = (−1)^(n(m−1)^(n−1))ψ(0).
10 Inequalities of the E-Determinant
The paper extends several matrix determinant inequalities to tensor E-determinants, including Hadamard-, Gershgorin-, and Minkowski-type results. The Hadamard-type bounds require positive semidefiniteness and stated spectral assumptions.
- Inequality extensions: The section generalizes several determinant inequalities for matrices to E-determinants of tensors.The stated extensions include Hadamard-, Gershgorin-, and a partial Minkowski-type inequality.
- Hadamard inequality: For even m, the tensor Hadamard inequality is developed under the section’s assumptions.When m = 2, the theorem reduces to the classical Hadamard inequality for matrices.
- Hadamard inequality: A sub-tensor U of a positive semidefinite symmetric tensor T has a nonnegative E-determinant.This lemma supplies a positivity property used in the inequality analysis.
- Hadamard inequality: If T is positive semidefinite and its diagonal entries satisfy t_i...i ≤ 1, then 0 ≤ Edet(T) ≤ 1 under Assumptions 8.1 and 10.1.The lower bound follows from positivity, while the upper bound is established using eigenvalue and arithmetic–geometric-mean arguments.
- Gershgorin inequality: The tensor analogue of Gershgorin’s inequality is formulated through the spectral radius ρ(T) = max_{λ∈σ(T)} |λ|.The proof proceeds by scaling tensor entries and applying the spectral-radius bound.
- Minkowski inequality: The paper presents a partial generalization of Minkowski’s matrix inequality to tensors.The result assumes even m, positive semidefiniteness, and Assumptions 8.1 and 10.1.
11 Final Remarks
The paper presents the E-determinant theory as applicable, while identifying unresolved questions about composition, assumptions, characteristic-polynomial formulas, and further determinant properties.
- The authors conclude that the E-determinant theory is applicable and worth further exploration.
- Whether Conjecture 4.1 holds, and what composition formula applies if it fails, remain open questions.
- Removing Assumption 8.1 and related assumptions, or characterizing tensors satisfying them, is identified as future work.
- More explicit characteristic-polynomial formulas for general tensors are needed beyond the dimension-two results.
- Further E-determinant properties, including a possible Laplace-type formula, could enable additional inequalities such as Oppenheim’s inequality.