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.
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