# Communications in Mathematical Sciences

## Volume 15 (2017)

### Inverse eigenvalue problem for tensors

Pages: 1627 – 1649

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n6.a7

#### Authors

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

Shenglong Hu (School of Mathematics, Tianjin University, Tianjin, China; and Department of Statistics, University of Chicago, Illinois, U.S.A.)

#### Abstract

Let $\mathbb{T}(\mathbb{C}^n , m+1)$ be the space of tensors of order m+1 and dimension n with complex entries. A tensor $\mathcal{T} \in \mathbb{T}(\mathbb{C}^n , m+1)$ has $nm^{n-1}$ eigenvalues (counted with algebraic multiplicities). The inverse eigenvalue problem for tensors is a generalization of the inverse eigenvalue problem for matrices. Namely, given a multiset $S \in \mathbb{C}^{nm^{n-1}} / \mathfrak{S} (nm^{n-1})$ of total multiplicity $nm^{n-1}$, is there a tensor in $\mathbb{T}(\mathbb{C}^n , m+1)$ such that the set of eigenvalues of $\mathcal{T}$ is exactly $S$? The solvability of the inverse eigenvalue problem for tensors is studied in this article. With tools from algebraic geometry, it is proved that the necessary and sufficient condition for this inverse problem to be generically solvable is $m=1$, or $n=2$, or $(n,m) = (3,2), (4,2), (3,3)$.

#### Keywords

tensor, eigenvalue, inverse problem

#### 2010 Mathematics Subject Classification

15A18, 15A69, 65F18