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.

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

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) $W(A+B)\subseteq W(A)+W(B)$, where the sum on the right-hand side denotes a sumset.
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. $r(A)\leq \|A\|\leq 2r(A)$, where $\|\cdot \|$ denotes the operator norm.
14. $r(A^{n})\leq r(A)^{n}$

Generalisations

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

See also

• Spectral theory
• Rayleigh quotient
• Workshop on Numerical Ranges and Numerical Radii

References

Bibliography

• Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys., 58: 77, arXiv:quant-ph/0511101, Bibcode:2006RpMP...58...77C, doi:10.1016/S0034-4877(06)80041-8.
• Dirr, G.; Helmkel, U.; Kleinsteuber, M.; Schulte-Herbrüggen, Th. (2006), "A new type of C-numerical range arising in quantum computing", Proc. Appl. Math. Mech., 6: 711–712.
• Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press, ISBN 978-0-521-07988-4.
• Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges II, Cambridge University Press, ISBN 978-0-521-20227-5.
• Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0-521-46713-1.
• Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc., 124: 1985.
• Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices", Linear Algebra and Its Applications, 252: 115.
• 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.