2014/12/09 by Shenglong Hu, Ke Ye, Hu, Shenglong +1
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Tensor decomposition and applications #math.SP
paper · pdf · doi:10.48550/arxiv.1412.2831
arxiv created 2014/12/09 · openalex publication_date 2014/12/09 · arxiv updated 2014/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.