The numerical range of a quantity, in ordinary usage, means the set of numerical values it can assume.
For a complex matrix 
of size 
, the numerical range is
the set
 |
where 
denotes the conjugate transpose of 
. It is a compact set
in the complex plane and contains every eigenvalue
of 
.
The numerical range is convex. If 
is a normal matrix, it is
the convex hull of the eigenvalues.
In general it can be larger. For example, the nilpotent
matrix ![[0 1; 0 0]](/images/equations/NumericalRange/Inline7.svg)
has only the eigenvalue 0 but its numerical range
is the closed disk of radius 
centered at 0.
The complete version of Crouzeix's conjecture compares operator norms of matrix-valued polynomials evaluated at an operator with their largest norm on its numerical range.
-Spectral
Set." SIAM J. Matrix Anal. Appl. 38, 649-655, 2017.