Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex [math]\displaystyle{ n \times n }[/math] matrix A is the set

[math]\displaystyle{ W(A) = \left\{\frac{\mathbf{x}^*A\mathbf{x}}{\mathbf{x}^*\mathbf{x}} \mid \mathbf{x}\in\mathbb{C}^n,\ \mathbf{x}\not=0\right\} }[/math]

where [math]\displaystyle{ \mathbf{x}^* }[/math] denotes the conjugate transpose of the vector [math]\displaystyle{ \mathbf{x} }[/math]. 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).

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.

[math]\displaystyle{ r(A) = \sup \{ |\lambda| : \lambda \in W(A) \} = \sup_{\|x\|=1} |\langle Ax, x \rangle|. }[/math]

Properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
3. [math]\displaystyle{ W(\alpha A+\beta I)=\alpha W(A)+\{\beta\} }[/math] for all square matrix [math]\displaystyle{ A }[/math] and complex numbers [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. Here [math]\displaystyle{ I }[/math] is the identity matrix.
4. [math]\displaystyle{ W(A) }[/math] is a subset of the closed right half-plane if and only if [math]\displaystyle{ A+A^* }[/math] is positive semidefinite.
5. The numerical range [math]\displaystyle{ W(\cdot) }[/math] is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. (Sub-additive) [math]\displaystyle{ W(A+B)\subseteq W(A)+W(B) }[/math], where the sum on the right-hand side denotes a sumset.
7. [math]\displaystyle{ W(A) }[/math] contains all the eigenvalues of [math]\displaystyle{ A }[/math].
8. The numerical range of a [math]\displaystyle{ 2 \times 2 }[/math] matrix is a filled ellipse.
9. [math]\displaystyle{ W(A) }[/math] is a real line segment [math]\displaystyle{ [\alpha, \beta] }[/math] if and only if [math]\displaystyle{ A }[/math] is a Hermitian matrix with its smallest and the largest eigenvalues being [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math].
10. If [math]\displaystyle{ A }[/math] is a normal matrix then [math]\displaystyle{ W(A) }[/math] is the convex hull of its eigenvalues.
11. If [math]\displaystyle{ \alpha }[/math] is a sharp point on the boundary of [math]\displaystyle{ W(A) }[/math], then [math]\displaystyle{ \alpha }[/math] is a normal eigenvalue of [math]\displaystyle{ A }[/math].
12. [math]\displaystyle{ r(\cdot) }[/math] is a norm on the space of [math]\displaystyle{ n \times n }[/math] matrices.
13. [math]\displaystyle{ r(A) \leq \|A\| \leq 2r(A) }[/math], where [math]\displaystyle{ \|\cdot\| }[/math] denotes the operator norm.
14. [math]\displaystyle{ r(A^n) \le r(A)^n }[/math]

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

Bibliography

• Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys. 58 (1): 77–91, doi:10.1016/S0034-4877(06)80041-8, Bibcode: 2006RpMP...58...77C .
• 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, doi:10.1002/pamm.200610336 .
• 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, Chapter 1, ISBN 978-0-521-46713-1 .
• Horn, Roger A.; Johnson, Charles R. (1990), Matrix Analysis, Cambridge University Press, Ch. 5.7, ex. 21, ISBN 0-521-30586-1 
• Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc. 124 (7): 1985, doi:10.1090/S0002-9939-96-03307-2 .
• Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices", Linear Algebra and Its Applications 252 (1–3): 115, doi:10.1016/0024-3795(95)00674-5 .
• "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.