Inverse eigenvalue problem for tensors

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

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Received 27 September 2016

Published 27 June 2017