Proof of Böttcher and Wenzel’s conjecture on commutator norms for 3-by-3 matrices

Böttcher and Wenzel conjectured that the Frobenius norm of the commutator of two real matrices of Frobenius norm 1 is at most and proved this for 2-by-2 matrices. In this paper, the conjecture is proved for 3-by-3 matrices.