Thanks, Christian Remling! There's a much harder question on MathOverflow to which Robin Chapman gave an answer to my question here.

In short: Bruce's guess is indeed the best we can do. This fact was proved by Schur, and there's also a proof here:

• Maryam Mirzakhani, A simple proof of a theorem of Schur, Amer. Math. Monthly 105 (1998), 260–262.

Thanks, Christian Remling! There's a much harder question on MathOverflow to which Robin Chapman gave an answer to my question here.

In short: Bruce's guess is indeed the best we can do. This fact was proved by Schur, and there's also a proof here:

• Maryam Mirzakhani, A simple proof of a theorem of Schur, Amer. Math. Monthly 105 (1998), 260–262.