---
title: "Crouzeix's conjecture"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Crouzeix's_conjecture"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:02:53.424444+00:00"
instance: "kb-cron"
---

Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows:

  
    
      
        ‖
        f
        (
        A
        )
        ‖
        ≤
        2
        
          sup
          
            z
            ∈
            W
            (
            A
            )
          
        
        
          |
        
        f
        (
        z
        )
        
          |
        
        ,
      
    
    {\displaystyle \|f(A)\|\leq 2\sup _{z\in W(A)}|f(z)|,}
  

where the set 
  
    
      
        W
        (
        A
        )
      
    
    {\displaystyle W(A)}
  
 is the field of values of a n×n (i.e. square) complex matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
  and 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a complex function that is analytic in the interior of 
  
    
      
        W
        (
        A
        )
      
    
    {\displaystyle W(A)}
  
 and continuous up to the boundary of 
  
    
      
        W
        (
        A
        )
      
    
    {\displaystyle W(A)}
  
.  Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and all complex polynomials 
  
    
      
        p
      
    
    {\displaystyle p}
  
:

  
    
      
        ‖
        p
        (
        A
        )
        ‖
        ≤
        2
        
          sup
          
            z
            ∈
            W
            (
            A
            )
          
        
        
          |
        
        p
        (
        z
        )
        
          |
        
      
    
    {\displaystyle \|p(A)\|\leq 2\sup _{z\in W(A)}|p(z)|}
  

holds, where the norm on the left-hand side is the spectral operator 2-norm.


== History ==
Crouzeix's theorem, proved in 2007, states that:

  
    
      
        ‖
        f
        (
        A
        )
        ‖
        ≤
        11.08
        
          sup
          
            z
            ∈
            W
            (
            A
            )
          
        
        
          |
        
        f
        (
        z
        )
        
          |
        
      
    
    {\displaystyle \|f(A)\|\leq 11.08\sup _{z\in W(A)}|f(z)|}
  

(the constant 
  
    
      
        11.08
      
    
    {\displaystyle 11.08}
  
  is independent of the matrix dimension, thus transferable to infinite-dimensional settings).
Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for 
  
    
      
        1
        +
        
          
            2
          
        
      
    
    {\displaystyle 1+{\sqrt {2}}}
  
, improving the original constant of 
  
    
      
        11.08
      
    
    {\displaystyle 11.08}
  
. More recently, dimension-dependent improvements have been obtained: Malman, Mashreghi, O'Loughlin and Ransford showed that for each fixed dimension 
  
    
      
        N
      
    
    {\displaystyle N}
  
 there exists a constant 
  
    
      
        
          C
          
            N
          
        
        <
        1
        +
        
          
            2
          
        
      
    
    {\displaystyle C_{N}<1+{\sqrt {2}}}
  
 such that the inequality holds for all 
  
    
      
        N
        ×
        N
      
    
    {\displaystyle N\times N}
  
 matrices.
Related work connects the constant in Crouzeix-type inequalities to configuration constants arising from the Neumann–Poincaré operator and yields domain-dependent improvements of the Crouzeix–Palencia bound in certain settings.

The not yet proved conjecture states that the constant can be refined to 
  
    
      
        2
      
    
    {\displaystyle 2}
  
.


== Special cases ==
While the general case is unknown, it is known that the conjecture holds for some special cases. For instance, it holds for all normal matrices, for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue and for general n×n matrices that are nearly Jordan blocks..  Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.


== Further reading ==
Ransford, Thomas; Schwenninger, Felix L. (2018-03-01). "Remarks on the Crouzeix–Palencia Proof that the Numerical Range is a 
  
    
      
        (
        1
        +
        
          
            2
          
        
        )
      
    
    {\displaystyle (1+{\sqrt {2}})}
  
-Spectral Set". SIAM Journal on Matrix Analysis and Applications. 39 (1): 342–345. arXiv:1708.08633. doi:10.1137/17M1143757. S2CID 43945191.
Gorkin, Pamela; Bickel, Kelly (2018-10-27). "Numerical Range and Compressions of the Shift". arXiv:1810.11680 [math.FA].


== References ==


== See also ==
Von Neumann's inequality