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Numerical range

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

  1. The numerical range is the range of the Rayleigh quotient.
  2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
  3. for all square matrix and complex numbers and . Here 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 (2), (3) and (4).
  6. for any unitary .
  7. .
  8. If is Hermitian, then is on the real line. If is anti-Hermitian, then is on the imaginary line.
  9. if and only if .
  10. (Sub-additive) .
  11. contains all the eigenvalues of .
  12. The numerical range of a matrix is a filled ellipse.
  13. is a real line segment if and only if is a Hermitian matrix with its smallest and the largest eigenvalues being and .

Normal matrices

  1. If is normal, and , where are eigenvectors of corresponding to , respectively, then .
  2. If is a normal matrix then is the convex hull of its eigenvalues.
  3. If is a sharp point on the boundary of , then is a normal eigenvalue of .

Numerical radius

  1. is a unitarily invariant norm on the space of matrices.
  2. , where denotes the operator norm.
  3. if (but not only if) is normal.
  4. .

Proofs

Most of the claims are obvious. Some are not.

General properties

The following proof is due to

Normal matrices

Numerical radius

Generalisations

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

Bibliography

Books

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Papers

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References