Resolve the symmetric-maximizer conjecture in dimension six

Determine whether the symmetric-maximizer conjecture for the Lyapunov operator is true or false for real matrices of order six; equivalently, determine whether every real 6 × 6 matrix A satisfies ||L_A|_{K_6}|| ≤ ||L_A|_{S_6}||.

Background

The Lyapunov operator associated with a real matrix A is defined by L_A(X) = AX + XAᵀ and is considered with the Frobenius-induced operator norm. The spaces S_n of symmetric matrices and K_n of skew-symmetric matrices are invariant under L_A, so the full operator norm is the larger of the norms of the two restrictions. The symmetric-maximizer conjecture asserts that the skew-symmetric restricted norm never exceeds the symmetric restricted norm, or equivalently that a norm-maximizing matrix can always be chosen symmetric.

The paper disproves this conjecture for order seven and, by a direct-sum construction, for every order n ≥ 7. Since the conjecture was previously proved for n ≤ 5, order six is the only dimension whose status remains unresolved in the paper.