Crouzeix's conjecture is an unsolved problem in matrix analysis.
It was proposed by Michel Crouzeix in 2004,[1] and it can be stated as follows: where the set
Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices
: holds, where the norm on the left-hand side is the spectral operator 2-norm.
Crouzeix's theorem, proved in 2007, states that:[2] (the constant
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
The not yet proved conjecture states that the constant can be refined to
[4] Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.