Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Multiplicities of tensor eigenvalues

Pages: 1049 – 1071

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a9

Authors

Shenglong Hu (Department of Mathematics, School of Science, Tianjin University, Tianjin, China; and Department of Mathematics, National University of Singapore)

Ke Ye (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Abstract

We study in this article multiplicities of tensor eigenvalues. There are two natural multiplicities associated to an eigenvalue of a tensor: algebraic multiplicity and geometric multiplicity. 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 in studying 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 an inequality relating the algebraic multiplicity and geometric multiplicity. We show that it is true for several cases, especially when the eigenvariety contains a linear subspace of dimension the geometric multiplicity of the given eigenvalue 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 multiplicity is not smaller than the geometric multiplicity.

Keywords

tensor, eigenvalue, eigenvector, algebraic multiplicity, geometric multiplicity

2010 Mathematics Subject Classification

15A18, 15A42, 15A69

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