Crouzeix's conjecture is a problem in matrix analysis proposed by the mathematician Michel Crouzeix in 2004. It concerns a bound relating the norm of a matrix function to the supremum of that function over the matrix's numerical range, or field of values.
Facts
Partially Attested
Progress Toward ResolutionOn July 27, 2026, neurosurgeon Shanmu Jin posted a preprint claiming a proof of Crouzeix's conjecture. 1 A bound with constant 11.08 was proven in 2007 (Crouzeix's theorem); the constant was later improved by Crouzeix and Palencia in 2017. Preprints claiming full proofs appeared in July and August 2026 and were reported as reviewed and believed correct by other specialists, but had not yet gone through full peer-reviewed journal publication as of the source date. StatementFor a square complex matrix A and an analytic complex function f, the norm of f(A) is at most twice the supremum of the absolute value of f over the field of values of A. 1 Proposed Year Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Crouzeix's Conjecture (Wikipedia)
Lead section, formal statement
proposed by Michel Crouzeix in 2004
Lead section
Crouzeix's conjecture is a problem in matrix analysis proposed by Michel Crouzeix in 2004
Proof section
On July 27, 2026, neurosurgeon Shanmu Jin posted a preprint claiming a proof of Crouzeix's conjecture.
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