Source-linked AI summary
The Number of Eigenvalues of a Tensor
Dustin Cartwright, Bernd Sturmfels
TL;DR
The paper asks how tensor eigenvalues and eigenvectors should be counted and interpreted beyond ordinary matrices. It defines the relevant equivalence classes, proves their generic number using toric geometry, and studies characteristic polynomials, projective dynamics, and symmetric tensors. The main count is ((m−1)n−1)/(m−2), while normalized eigenvalues of symmetric tensors remain finite, subject to stated scope boundaries.
Problem
Tensor eigenvalues and eigenvectors extend familiar linear-algebra concepts to multilinear algebra, but their counting and normalization require a suitable framework.
Method
The paper models tensor eigenpairs as fixed points or weighted-projective intersection solutions and analyzes characteristic polynomials and symmetric tensors.
Results
((m−1)n−1)/(m−2) is the generic number of tensor eigenpair equivalence classes, and a generic tensor’s characteristic polynomial has that degree and is irreducible.
Takeaways & Limitations
Eigenpair counting extends the matrix result to higher-order tensors, while symmetric tensors have finitely many normalized eigenvalues bounded by the same quantity.
Takeaways & Limitations
For n≥3, zero can be the only eigenvalue even when the associated polynomials have no common factors and the tensor is not nilpotent.
Abstract
from arXiv · showhide
Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the number of normalized eigenvalues of a symmetric tensor is always finite. We also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants.
1. Introduction
The paper extends tensor eigenvalue concepts from multilinear algebra and counts eigenpair equivalence classes for generic tensors. It also introduces normalization choices and connects the count to characteristic polynomials, projective dynamics, and symmetric tensors.
- Motivation: Tensor eigenvalues are defined through E-eigenvalues, extending a matrix concept that has applications across numerical multilinear algebra.The paper adopts Qi’s definition and uses “eigenvalue” to mean E-eigenvalue throughout.
- Normalization: Normalization choices matter: Qi’s condition x·x=1 excludes eigenvectors with x·x=0, so it does not strictly generalize classical matrix eigenvalues.The paper calls eigenvalues with an eigenvector satisfying x·x=1 normalized eigenvalues.
- Eigenpair equivalence: Eigenpairs are treated modulo the scaling equivalence (λ,x) ~ (t^(m−2)λ,tx), avoiding a normalization choice in most of the paper.This generalizes the matrix ambiguity that eigenvectors are defined only up to scaling.
- Main count: ((m−1)n−1)/(m−2) equivalence classes is the generic number of tensor eigenpairs over C, counted without multiplicity when entries are sufficiently generic.The theorem counts multiplicities generally; sufficiently generic tensors have multiplicity one for every class.
- Proof strategy: The paper gives a short toric-geometry proof of a formula previously established for even-order tensors and conjectured for arbitrary order.The earlier even-order derivation used a Macaulay matrix for the multivariate resultant.
- Further directions: The paper also studies characteristic polynomials, projective fixed-point dynamics, and the finiteness of normalized eigenvalues for symmetric tensors.Its organization includes characteristic-polynomial and singular-tensor analysis, projective dynamics, and symmetric tensors.
2. Intersections in a Weighted Projective Space
The eigenpair problem is recast as an intersection problem in weighted projective space. Toric intersection theory yields the exact generic count, while examples establish multiplicity, sharpness, and real-eigenpair consequences.
- Weighted projective formulation: The eigenvalue problem is formulated as the intersection of homogeneous polynomial equations in an n-dimensional weighted projective space.The coordinates rescale as (tu_1:···:tu_n:t^(m−2)λ), assigning λ weighted degree m−2.
- Toric setup: The toric variety is built from an n-dimensional simplex whose normalized volume is (m−2)^(n−1).The simplex is associated with the very ample line bundle O_X(m−2).
- Intersection count: The intersection computation removes the trivial solution x=0, which contributes 1/(m−2) because it corresponds to the singular weighted-projective point.After this correction, the number of non-trivial solutions equals the theorem’s positive integer count.
- Intersection count: For tensors with finitely many eigenpair classes, the multiplicity-counted number is ((m−1)n−1)/(m−2).A diagonal example attains this bound with every solution having multiplicity one, implying the same multiplicity behavior for generic tensors.
- Alternative count: A substitution λ=˜λ^(m−2), followed by Bézout counting and orbit identification, independently explains the same quotient by m−2.The generic projective system has (m−1)^n−1 nontrivial solutions before grouping the root-of-unity orbits.
- Examples: Diagonal tensors explicitly realize the count, and for generic diagonal coefficients all ((m−1)n−1)/(m−2) eigenpairs have distinct normalized eigenvalues.The construction uses roots of unity to enumerate representatives and then groups equivalent eigenpairs.
- Real eigenpairs: If A has real entries and either m or n is odd, it has a real eigenpair; this condition is sharp because even m and n can yield no real eigenpairs.The paper constructs even-dimensional real tensors without real eigenpairs using block-diagonal copies of a two-dimensional example.
3. Characteristic Polynomial and Singular Tensors
The characteristic polynomial is obtained by eliminating eigenvector variables, and its degree gives the number of normalized eigenvalues for a generic tensor. The paper distinguishes several singularity notions and exhibits tensors whose normalized-eigenvalue sets are finite, cofinite, or otherwise exceptional.
- The characteristic polynomial is formed by eliminating x1, . . . , xn from Ax^(m−1) = λx and x · x = 1; for odd m, the resulting polynomial has the form φA(λ^2).
- For a generic tensor, the characteristic polynomial is irreducible and has degree ((m −1)n −1)/(m −2), which equals the number of normalized eigenvalues.
- A complex 2 × 2 × 2 tensor has every nonzero complex number as a normalized eigenvalue, but zero is not normalized, providing a set that is neither finite nor all of C.
- The set of normalized eigenvalues is either finite or the complement of a finite set, because it is a constructible subset of C.
- The implications from all normalized eigenvalues to infinitude and identically vanishing characteristic polynomial are strict: an identically vanishing polynomial need not yield infinitely many normalized eigenvalues.
- For 2 × 2 × 2 tensors, the singular variety is irreducible of codimension 2 and degree 4, while its real locus is the union of two codimension-4 linear spaces.
4. Dynamics on Projective Space
The paper interprets tensor eigenvectors as fixed points of a rational self-map on projective space, with zero eigenvalues corresponding to its base locus. It shows that tensor nilpotence implies zero is the only eigenvalue, but the converse fails.
- Tensor eigenpairs and projective dynamics: The paper proposes projective dynamics as a potentially useful tool for modeling and numerical computations involving tensor eigenpairs.
- Tensor eigenpairs and projective dynamics: Tensor eigenvectors with non-zero eigenvalue are exactly the fixed points of ψA on P^(n−1), while zero-eigenvalue eigenvectors form its base locus.Thus ψA is defined everywhere exactly when 0 is not an eigenvalue.
- Nilpotence: Nilpotence implies that 0 is the only eigenvalue, but tensors with only eigenvalue 0 need not be nilpotent.This generalizes the classical implication while explicitly rejecting its converse for tensors.
- Nilpotence: For n = 3, tensors can have zero as their only eigenvalue, no common polynomial factors in Ax^(m−1), and still fail to be nilpotent.
- Examples: In the m = n = 3 Cremona example, the only eigenvectors up to scaling are three base-locus points, all with eigenvalue 0, although the map is not nilpotent.
5. Symmetric Tensors
For symmetric tensors, eigenvectors and normalized eigenvalues admit polynomial and geometric characterizations. The paper proves finiteness and a sharp generic bound for normalized eigenvalues, while relating the characteristic polynomial to discriminants with possible extraneous factors.
- Polynomial representation: A symmetric tensor corresponds uniquely to a homogeneous polynomial, and its tensor contraction gives the polynomial gradient coordinates.This polynomial viewpoint underlies the section’s geometric analysis of eigenvectors.
- Subtleties and applications: Eigenvalues of symmetric tensors can test positive semidefiniteness of real even-degree polynomials, and the Motzkin example has four distinct real eigenvalues.For the associated degree-six tensor, there are 25 eigenvalues counting multiplicities, compared with the upper bound 31.
- Polynomial representation: Zero-eigenvalue eigenvectors are precisely the singular points of the projective hypersurface defined by the corresponding polynomial.
- Polynomial representation: For nonzero λ and m ≥ 3, normalized eigenvectors with eigenvalue λ are precisely singular points of a related affine hypersurface.
- Characteristic polynomial: The characteristic polynomial is a factor of the multivariate discriminant, but the discriminant may also contain other irreducible factors.For n = 2 and m = 3, the discriminant equals λ4·φA(λ); the quartic example likewise has an extraneous binary-quartic factor, and arbitrary factorization remains open.
- Finiteness and bound: Every symmetric tensor has finitely many normalized eigenvalues, because normalized eigenvalues correspond to critical values of a polynomial restricted to a smooth variety.The proof uses generic smoothness, equivalently Sard’s theorem, together with the finite bound on critical values in this setting.
- Finiteness and bound: ((m −1)n −1)/(m −2) is an upper bound on the number of distinct normalized eigenvalues, and generic symmetric tensors attain it.The argument bounds connected components of the eigenpair set through an intersection number of ample divisors on a weighted projective space.
- Subtleties and applications: The normalized-eigenvalue theorem depends intrinsically on x · x = 1 and does not ensure that every symmetric tensor has a non-trivial characteristic polynomial.One example has only normalized eigenvalue 1 while its characteristic polynomial is identically zero; an alternative normalization can yield infinitely many eigenvalue magnitudes.