Inverse tensor eigenvalue problem

TL;DR

This paper investigates the inverse eigenvalue problem for tensors, establishing conditions under which a tensor with a prescribed eigenvalue multiset exists, using algebraic geometry tools to identify when the problem is generically solvable.

Contribution

It provides a complete characterization of the conditions for the generic solvability of the inverse tensor eigenvalue problem.

Findings

The inverse eigenvalue problem is solvable if and only if m=1, or n=2, or (n,m) is (3,2), (4,2), or (3,3).

Algebraic geometry techniques are used to analyze the problem.

The paper extends classical eigenvalue problems from matrices to tensors.

Abstract

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