Spaces of matrices with few eigenvalues

TL;DR

This paper investigates the structure and maximal dimension of matrix subspaces over a field where matrices have at most two eigenvalues, providing bounds and classifications for these spaces.

Contribution

It establishes upper bounds on the dimension of such matrix subspaces and classifies the maximal spaces up to similarity.

Findings

Dimension of subspaces with at most two eigenvalues is ≤ n(n-1)/2.

Maximal subspaces are classified up to similarity.

Results hold for fields with characteristic not 2.

Abstract

Let K be a (commutative) field with characteristic not 2, and V be a linear subspace of n by n matrices that have at most two eigenvalues in K (respectively, at most one non-zero eigenvalue in K). We prove that the dimension of V is less than or equal to n(n-1)/2 provided that n be greater than 2 (respectively, the dimension of V is less than or equal to n(n-1)/2). We also classify, up to similarity, the linear subspaces of n by n matrices in which every matrix has at most two eigenvalues (respectively, at most one non-zero eigenvalue) in an algebraic closure of K and which have the maximal dimension among such spaces.