In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex
denotes the conjugate transpose of the vector
The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors).
In engineering, numerical ranges are used as a rough estimate of eigenvalues of A.
Recently, generalizations of the numerical range are used to study quantum computing.
A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e. Let sum of sets denote a sumset.
General properties Normal matrices Numerical radius Most of the claims are obvious.
is Hermitian, then it is normal, so it is the convex hull of its eigenvalues, which are all real.
would stray from the real line.
The space of all one-dimensional subspaces of
The image of a 2-sphere under a linear projection is a filled ellipse.
is the composition of two real linear maps
, which maps the 2-sphere to a filled ellipse.
is the image of a continuous map
from the closed unit sphere, so it is compact.
of unit norm, project
is a filled ellipse by the previous result, and so for any
be the original numerical range.
, then we can translate and rotate the complex plane so that the point translates to the origin, and the region
and that inequality is sharp, meaning that
This is a complete characterization of the supporting planes of
is normal, then it has a full eigenbasis, so it reduces to (1).
is normal, by the spectral theorem, there exists a unitary matrix
is a diagonal matrix containing the eigenvalues
, we can translate and rotate the complex plane, so that we reduce to the case where
, and that the two supporting planes at that point both make an angle
, there exists a unit vector
By general property (4), the numerical range lies in the sectors defined by:
must vanish to maintain non-negativity.
is on the imaginary line, the extremal points of