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Crouzeix's Conjecture

Crouzeix's conjecture (Crouzeix 2004) is the assertion that every complex matrix A of size n×n and every polynomial p satisfy

||p(A)||_2<=2max_(z in W(A))|p(z)|,

where W(A) is the numerical range and ||·||_2 is the spectral norm. The conjectured constant 2 is sharp. For the nilpotent matrix A=[0 1; 0 0] and p(z)=z, the spectral norm is 1, whereas the numerical range is the closed disk of radius 1/2 centered at 0, so equality holds and no smaller universal constant is possible.

The best general constant was reduced to 1+sqrt(2) by Crouzeix and Palencia (2017). Jin (2026) gave a proof with the sharp constant 2, with the key argument generated by GPT-5.6 Sol. Townsend and Greenbaum (2026) report that they and Crouzeix independently checked Jin's proof, and Lorist and Schwenninger (2026) subsequently gave another proof.

See also

Matrix Polynomial, Numerical Range, Spectral Norm

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References

Crouzeix, M. "Bounds for Analytical Functions of Matrices." Integral Equations and Operator Theory 48, 461-477, 2004. https://doi.org/10.1007/s00020-002-1188-6.Crouzeix, M. and Palencia, C. "The Numerical Range Is a (1+sqrt(2))-Spectral Set." SIAM J. Matrix Anal. Appl. 38, 649-655, 2017. https://doi.org/10.1137/17M1116672.Jin, S. "The Numerical Range Is a 2-Spectral Set." July 27, 2026. https://www.preprints.org/manuscript/202607.1919.Lorist, E. and Schwenninger, F. "A Solution to Crouzeix's Conjecture." 4 Aug 2026. https://arxiv.org/abs/2608.03841.Townsend, A. and Greenbaum, A. "The Neurosurgery Resident Who Proved Crouzeix's Conjecture." Aug. 15, 2026. https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf.

Cite this as:

Weisstein, Eric W. "Crouzeix's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CrouzeixsConjecture.html

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