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Böttcher-Wenzel Inequality

The Böttcher-Wenzel inequality bounds the Frobenius norm of a matrix commutator. For complex n×n matrices,

||XY-YX||_F^2<=2||X||_F^2||Y||_F^2.

The constant 2 is optimal for n>=2 (Fox 2026).

Fox (2026) proved a sharp extension to a complex simple Lie algebra g. Let tau be conjugation with respect to a compact real form, let B be the Killing form, and set H(x,y)=-B(x,tauy) and ||x||^2=H(x,x). Then

||[x,y]||^2<=1/(h^ v )||x||^2||y||^2,

where h^ v is the dual Coxeter number. For nonzero x and y, equality holds precisely when they are H-orthogonal and contained in a tau-stable long-root subalgebra isomorphic to sl(2,C).

See also

Commutator, Dual Coxeter Number, Frobenius Norm, Killing Form, Lie Algebra

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References

Fox, D. J. F. "Sharp Commutator Bounds on Complex Simple Lie Algebras: The Böttcher-Wenzel Inequality and the Comass of the Cartan 3-Form." 8 Oct 2026. https://arxiv.org/abs/2610.11800.

Cite this as:

Weisstein, Eric W. "Böttcher-Wenzel Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bottcher-WenzelInequality.html

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