Spaces of matrices with a sole eigenvalue

Abstract

Let 𝕂 be an arbitrary (commutative) field and be an algebraic closure of it. Let V be a linear subspace of M _n (𝕂), with n ≥ 3. We show that if every matrix of V has at most one eigenvalue in 𝕂, then . If every matrix of V has a sole eigenvalue in and , we show that V is similar to the space of all upper-triangular matrices with equal diagonal entries, except if n = 3 and 𝕂 has characteristic 3, or if n = 4 and 𝕂 has characteristic 2. In both of those special cases, we classify the exceptional solutions up to similarity.

Keywords:

• linear subspaces
• spectrum
• nilpotent matrices
• Gerstenhaber theorem
• positive characteristic

AMS Subject Classification:

• Primary: 15A30
• Secondary: 15A18