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

$W(A) = \left\{\frac{\mathbf{x}^*A\mathbf{x}}{\mathbf{x}^*\mathbf{x}} \mid \mathbf{x}\in\mathbb{C}^n,\ x\not=0\right\}$

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.

$r(A) = \sup \{ |\lambda| : \lambda \in W(A) \} = \sup_{\|x\|=1} |\langle Ax, x \rangle|.$

r(A) is a norm. r(A)=||A||, where ||A|| is the operator norm of A.

Properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
3. $W(\alpha A+\beta I)=\alpha W(A)+\{\beta\}$ for all square matrix A and complex numbers α and β. Here I is the identity matrix.
4. $W(A)$ is a subset of the closed right half-plane if and only if $A+A^*$ is positive semidefinite.
5. The numerical range $W(\cdot)$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. (Sub-additive) $W(A+B)\subseteq W(A)+W(B)$.
7. $W(A)$ contains all the eigenvalues of A.
8. The numerical range of a 2×2 matrix is an elliptical disk.
9. $W(A)$ is a real line segment [α, β] 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 $W(A)$ is the convex hull of its eigenvalues.
11. If α is a sharp point on the boundary of $W(A)$, then α is a normal eigenvalue of A.
12. $r(\cdot)$ is a norm on the space of n×n matrices.
13. $r(A^n) \le r(A)^n$

Generalisations

• C - numerical range
• Higher rank numerical range
• Joint numerical range
• Product numerical range
• Polynomial numerical hull

See also

• Spectral theory
• Rayleigh quotient

References

Bibliography

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• Roger A. Horn and Charles R. Johnson, Topics in Matrix Analysis, Chapter 1, Cambridge University Press, 1991. ISBN 0-521-30587-X (hardback), ISBN 0-521-46713-6 (paperback).
• "Functional Characterizations of the Field of Values and the Convex Hull of the Spectrum", Charles R. Johnson, Proceedings of the American Mathematical Society, 61(2):201-204, Dec 1976.