Numerical range


In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n × n matrix A is the set
where x* denotes the conjugate transpose of the vector x.
In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of 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

  1. The numerical range is the range of the Rayleigh quotient.
  2. The numerical range is convex and compact.
  3. for all square matrix A and complex numbers α and β. Here I is the identity matrix.
  4. is a subset of the closed right half-plane if and only if is positive semidefinite.
  5. The numerical range is the only function on the set of square matrices that satisfies, and.
  6. , where the sum on the right-hand side denotes a sumset.
  7. contains all the eigenvalues of A.
  8. The numerical range of a 2×2 matrix is an elliptical disk.
  9. is a real line segment if and only if A is a Hermitian matrix with its smallest and the largest eigenvalues being α and β
  10. If A is a normal matrix then is the convex hull of its eigenvalues.
  11. If α is a sharp point on the boundary of, then α is a normal eigenvalue of A.
  12. is a norm on the space of n×n matrices.
  13. , where denotes the operator norm.

  14. Generalisations