Multiplicities of tensor eigenvalues

Abstract

We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue λ of a tensor: algebraic multiplicity am(λ) and geometric multiplicity gm(λ). The former is the multiplicity of the eigenvalue as a root of the characteristic polynomial, and the latter is the dimension of the eigenvariety (i.e., the set of eigenvectors) corresponding to the eigenvalue.We show that the algebraic multiplicity could change along the orbit of tensors by the orthogonal linear group action, while the geometric multiplicity of the zero eigenvalue is invariant under this action, which is the main difficulty to study their relationships. However, we show that for a generic tensor, every eigenvalue has a unique (up to scaling) eigenvector, and both the algebraic multiplicity and geometric multiplicity are one. In general, we suggest for an m-th order n-dimensional tensor the relationship am(λ) ≥ gm(λ)(m − 1) gm(λ)−1 . We show that it is true for serveral cases, especially when the eigenvariety contains a linear subspace of dimension gm(λ) in coordinate form. As both multiplicities are invariants under the orthogonal linear group action in the matrix counterpart, this generalizes the classical result for a matrix: the algebraic mutliplicity is not smaller than the geometric multiplicity.2010 Mathematics Subject Classification. 15A18; 15A42; 15A69.

Nondegeneracy of eigenvectors and singular vector tuples of tensors

“…However, for a given eigenvalue λ ∈ σ(A), the set of the corresponding eigenvectors V (λ) (adding the zero vector) is not a linear subspace of C n any more. It is an eigenvariety [18]. In general, the eigenvariety is rather complicated.…”

Eigenvariety of nonnegative symmetric weakly irreducible tensors associated with spectral radius and its application to hypergraphs

“…Li et.al [11] give a new definition of geometric multiplicity based on nonnegative irreducible tensors and two-dimensional nonnegative tensors. Hu and Ye [10] define the geometric multiplicity of an eigenvalue λ of A to be the dimension of V λ as an affine variety, and try to establish a relationship between the algebraic multiplicity and geometric multiplicity of an eigenvalue of A.…”

Inverse eigenvalue problem for tensors

“…As long as we are concerning on eigenvalues of tensors, which are solely related to T x m , it is sufficient to consider the tensor space TS(C n , m + 1) := C n ⊗ S m (C n ) (cf. [15,Section 5.2]). For any T ∈ T(C n , m + 1), we can symmetrize its ith slice T i :…”

“…Definition 2.1). However, both computation and structures of the eigenvalues are very complicated, and tough to investigate [14,15,24]. The situation would be improved if the eigenvalues are shown to lie in a variety in C nm n−1 with a much smaller dimension.…”