Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex $n\times n$ matrix A is the set

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

where $\mathbf {x} ^{*}$ denotes the conjugate transpose of the vector $\mathbf {x}$. 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.

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

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 $\alpha$ and $\beta$. 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 $2\times 2$ matrix is a filled ellipse.
9. $W(A)$ is a real line segment $[\alpha ,\beta ]$ if and only if $A$ is a Hermitian matrix with its smallest and the largest eigenvalues being $\alpha$ and $\beta$.
10. If $A$ is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
11. If $\alpha$ is a sharp point on the boundary of $W(A)$, then $\alpha$ is a normal eigenvalue of $A$.
12. $r(\cdot )$ is a norm on the space of $n\times n$ 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

Bibliography

• Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys., 58 (1): 77–91, arXiv:quant-ph/0511101, Bibcode:2006RpMP...58...77C, doi:10.1016/S0034-4877(06)80041-8, S2CID 119427312.
• 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.