Helly-type theorem for eigenvectors

Abstract

We prove that if any ⌊3d/2⌋ or fewer elements of a finite family of linear operators K d → K d (K is an arbitrary field) have a common eigenvector then all operators in the family have a common eigenvector. Moreover, ⌊3d/2⌋ cannot be replaced by a smaller number. Also, we study the following problem, achieving partial results: prove that if any l = O(d) or fewer elements of a finite family of linear operators K d → K d have a common non-trivial invariant subspace then all operators in the family have a common non-trivial invariant subspace.