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 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).

is a Hermitian projection operator 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

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. is the identity matrix.

4. is positive semidefinite.

5. is the only function on the set of square matrices that satisfies (2), (3) and (4).

6. .

7. .

8. is anti-Hermitian is on the imaginary line.

9. .

10. .

11. contains all the eigenvalues.

12. matrix is a filled ellipse.

13. is a Hermitian matrix.

Normal matrices

1. .

2. is the convex hull of its eigenvalues.

3. is a normal eigenvalue.

Numerical radius

1. is a unitarily invariant norm matrices.

2. denotes the operator norm.

3. 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

• C-numerical range

• Joint numerical range

• Product numerical range

• Polynomial numerical hull

Higher-rank numerical range

This allows a generalization to higher-rank numerical ranges: is always closed and convex,.

See also

• Spectral theory

• Rayleigh quotient

• Workshop on Numerical Ranges and Numerical Radii

Bibliography

Books

• .

Papers

• .

• .

• .

References