In the mathematical field of linear algebra and convex analysis, the numerical range or field of values or Wertvorrat or Wertevorrat of a complex matrix A is the set
where 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).
Equivalently, the elements of are of the form , where is a Hermitian projection operator from to a one-dimensional subspace.
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.
Properties Let sum of sets denote a sumset.
General properties The numerical range is the range of the Rayleigh quotient. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact. for all square matrix and complex numbers and . Here is the identity matrix. is a subset of the closed right half-plane if and only if is positive semidefinite. The numerical range is the only function on the set of square matrices that satisfies (2), (3) and (4). for any unitary . . If is Hermitian, then is on the real line. If is anti-Hermitian, then is on the imaginary line. if and only if . (Sub-additive) . contains all the eigenvalues of . The numerical range of a matrix is a filled ellipse. is a real line segment if and only if is a Hermitian matrix with its smallest and the largest eigenvalues being and . Normal matrices If is normal, and , where are eigenvectors of corresponding to , respectively, then . If is a normal matrix, then is the convex hull of its eigenvalues. If is a sharp point on the boundary of , then is a normal eigenvalue of . Numerical radius is a unitarily invariant norm on the space of matrices. , where denotes the operator norm. if (but not only if) is normal. .
Proofs Most of the claims are obvious. Some are not.
General properties The following proof is due to
Normal matrices
Numerical radius
Generalisations C-numerical range Joint numerical range Product numerical range Polynomial numerical hull
Higher-rank numerical range The numerical range is equivalent to the following definition:
This allows a generalization to higher-rank numerical ranges, one for each :
is always closed and convex, but it might be empty. It is guaranteed to be nonempty if , and there exists some such that is empty if .
See also Spectral theory Rayleigh quotient Workshop on Numerical Ranges and Numerical Radii
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