Tensors with eigenvectors in a given subspace

TL;DR

This paper extends the study of the Kalman variety from matrices to symmetric tensors, proving its irreducibility and computing its geometric properties, with new results on tensors with singular t-ples.

Contribution

It generalizes the Kalman variety to symmetric tensors, establishing irreducibility and calculating codimension and degree, including new results for tensors with singular t-ples.

Findings

Kalman variety is irreducible for symmetric tensors

Computed codimension and degree of the tensor Kalman variety

Proved analogous results for tensors with singular t-ples

Abstract

The first author with B. Sturmfels studied the variety of matrices with eigenvectors in a given linear subspace, called Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore we consider the Kalman variety of tensors having singular t-ples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.